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NCERT Solutions for Class 11 Maths Chapter 9 Straight Lines

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NCERT Solutions for Maths Class 11 Chapter 9 Straight Lines - FREE PDF Download

Class 11 Maths NCERT Solutions for Chapter 9 Straight Lines, provided by Vedantu. This chapter is fundamental as it introduces you to the concept of straight lines, covering essential topics like the slope of a line, different forms of the equation of a line, and the angle between two lines. Understanding these basics is crucial for solving complex problems in higher mathematics.

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In this chapter focus on key concepts such as the intercept form, point-slope form, and general equation of a line. Pay special attention to solving problems related to parallel and perpendicular lines, as they form the basis for many real-world applications. With Vedantu’s step-by-step solutions for Class 11 Maths Syllabus, you can understand these concepts easily and excel in your exams.


Access Exercise wise NCERT Solutions for Chapter 9 Maths Class 11

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Exercises Under NCERT Solutions for Class 11 Maths Chapter 9 Straight Lines

  • Exercise 9.1: This exercise introduces the concept of a straight line, its slope, and its intercepts. Students will learn to find the slope and intercepts of a line, as well as its equation in various forms.

  • Exercise 9.2: In this exercise, students will learn about the point-slope form of the equation of a line, the two-point form of the equation of a line, and the slope-intercept form of the equation of a line. They will practice using these forms to find the equation of a line given certain information.

  • Exercise 9.3: This exercise focuses on the distance and midpoint formula. Students will learn how to find the distance between two points and the midpoint of a line segment.

  • Miscellaneous Exercise: This exercise includes a mix of questions covering all the concepts taught in the chapter. Students will have to apply their knowledge of straight lines to solve various problems and answer questions. They will also practice finding the equation of a line and calculating its slope, intercepts, distance, and midpoint.


Access NCERT Solutions for Class 11 Maths Chapter 9 – Straight Lines

Exercise 9.1

1. Draw a quadrilateral in the Cartesian plane, whose vertices are (4,5),(0,7),(5,5)  and (4,2). Also, find its area.

Ans.Assume PQRS is the quadrilateral with given vertices P(4,5),Q(0,7),R(5,5), and S(4,2). 

Plot P, Q, R and S on the Cartesian plane and join PQ,QR,RS, and SP. Then draw the diagonal PR.


the vertices of a triangle are


From the figure, area (PQRS)= area (ΔPQR)+ area (ΔPRS) 

If the vertices of a triangle are (x1,y1),(x2,y2), and (x3,y3) , the area is,

12|x1(y2y3)+x2(y3y1)+x3(y1y2)|

Substitute the values and find the area of each triangle,

Now, area of ΔPQR =124(7+5)+0(55)+5(57) unit 2

=12|4(7+5)+0(55)+5(57)| unit 2

=12|4(12)+5(2)| unit 2

=12|4810| unit 2

=12|58| unit 2

=12×58 unit 2

=29 unit 2

Similarly area of ΔPRSis,

=12|4(5+2)+5(25)+(4)(55)| unit 2

=12|4(3)+5(7)4(10)| unit 2

=12|123540| unit 2

=12|63| unit 2

=632 unit 2

We have, area (PQRS)= area (ΔPQR)+ area (ΔPRS) 

=(29+632) unit 2

=58+632 unit 2

=1212 unit 2

Therefore, the area of the quadrilateral is 1212 unit 2.


2. The base of an equilateral triangle with side 2a lies along the y-axis such that the midpoint of the base is at the origin. Find vertices of the triangle.

Ans.PQR be the given equilateral triangle with side 2a

We know that all the sides of the equilateral triangle will be equal.

That is,

PQ=QR=RP=2a

Assume that base QR lies along the y-axis such that the mid-point of QR is at the origin.

That is, QO=OR=a, where O is the origin. 

Now, the coordinates of point R are (0,a), while the coordinates of point Qare (0,a).

The line joining a vertex of an equilateral triangle with the mid-point of its opposite side is perpendicular. 

Thus, the vertex Plies on the y-axis.

Now, plot the figure,


By applying Pythagoras theorem


By applying Pythagoras theorem to ΔPOR

(PR)2=(OP)2+(OR)2

Substitute the values,

(2a)2=(OP)2+a2

4a2a2=(OP)2

(OP)2=3a2

OP=3a

Thus, Coordinates of point P=(±3a,0) 

Therefore, the vertices of the equilateral triangle are (0,a),(0,a), and (3a,0)or(0,a),(0,a), and (3a,0).


3. Find the distance between P(x1,y1) and Q(x2,y2) when 

(i) PQis parallel to the y-axis

Ans.Here, the points are P(x1,y1) and Q(x2,y2).

When PQ is parallel to the y-axis, we have x1=x2

The distance between P and Qis,

(x2x1)2+(y2y1)2 

=(y2y1)2 (since x1=x2.)

=|y2y1|

Therefore, the distance between P(x1,y1) and Q(x2,y2) when PQis parallel to the y-

axis is |y2y1|.

(ii) PQ is parallel to the x-axis.

Ans.The points are P(x1,y1) and Q(x2,y2).

When PQ is parallel to the x-axis, we know y1=y2

Distance between P and Qis,

(x2x1)2+(y2y1)2

=(x2x1)2 (since y1=y2)

=|x2x1|

Therefore, the distance between P(x1,y1) and Q(x2,y2) when PQ is parallel to the x-

axis is |x2x1|.


4. Find a point on the x-axis, which is equidistant from the points (7,6) and (3,4).

Ans.Let A(7,6) and B(3,4)be the given points.

Assume C(a,0) as the point on the X - axis that is equidistance from the points (7,6) 

and (3,4).In general, Distance between two points is (x2x1)2+(y2y1)2

Now, Distance between A and C= Distance between B and C.

(7a)2+(60)2=(3a)2+(40)2

49+a214a+36=9+a26a+16  (since (ab)2=a22ab+b2)

a214a+85=a26a+25

Square both sides, 

a214a+85=a26a+25

14a+6a=2585

8a=60

a=608

 =152

Therefore, the point on the x-axis which is equidistant from the points (7,6) and (3,4)is 

(152,0).


5. Find the slope of a line, which passes through the origin, and the mid-point of the segment joining the points P(0,4) and B(8,0).

Ans.The coordinates of the midpoint of the line segment joining two points are,

(x1+x22,y1+y22)

Then, the coordinates of the mid-point of the line segment joining the points P(0,4) and B(8,0) are,

(0+82,4+02)=(4,2)

We know, the slope (m)of a non-vertical line passing through the points (x1,y1) and (x2,y2) is,

m=y2y1x2x1,where x2x1

Thus, the slope of the line passing through (0,0,, and (4,2) is,

 2040

 =24

 =12 

Therefore, the slope of a line, which passes through the origin, and the mid-point of the

segment joining the points P(0,4) and B(8,0) is 12.


6. Without using the Pythagoras theorem, show that the points (4,4),(3,5) and (1,1) are vertices of a right angled triangle.

Ans.The vertices of the given triangle are P(4,4),Q(3,5), and R(1,1).

We know, the slope (m)of a non-vertical line passing through the points (x1,y1) and (x2,y2) is,

m=y2y1x2x1,where x2x1

Now calculate the slope of each line,

Slope of PQ=5434=1

Slope of QR=1513=64=32

Slope of RP=4+14+1=55=1

Here, Slope of PQ= Slope of RP=1

This means that the line segments PQ and RP are perpendicular to each other

That is, the triangle is right-angled at P(4,4)

Therefore, the points (4,4),(3,5) and (1,1)are vertices of a right angled triangle.


7. Find the slope of the line, which makes an angle of 30 with the positive direction of y-axis measured anticlockwise.

Ans.Given that, the line makes an angle of 30 with the positive direction of y-axis 

measured anticlockwise.

Plot the figure,


he figure, the angle made by the line with the positive direction.


From the figure, the angle made by the line with the positive direction of the x-axis 

measured anticlockwise is,

 90+30=120

Now, the slope of the given line is tan120

Rewritetan120as tan(18060)

=tan60

=3

Therefore, the slope of the line, which makes an angle of 30 with the positive direction of y-axis measured anticlockwise is 3.


9. Without using distance formula, show that points (2,1),(4,0),(3,3) and (3,2) are vertices of a parallelogram.

Ans.Let points (2,1),(4,0),(3,3) and (3,2) be respectively denoted by P,Q,R and S.

Slopes of PQ=0+14+2=16

Slopes of RS=2333=16=16

Here, Slope of PQ= Slope of RS

This means, PQ and RS are parallel to each other.

Slope of QR=3034=31=3

Slope of PS=2+13+2=31=3

Here, Slope of QR= Slope of PS 

This means, QR and PS are parallel to each other. 

Thus, both pairs of opposite side of the quadrilateral PQRS are parallel and it is a 

parallelogram.

Therefore, (2,1),(4,0),(3,3) and (3,2) are vertices of a parallelogram.


9. Find the angle between the x-axis and the line joining the point

(3, 1) and (4, 2).

Ans.The points are (3, 1) and (4, 2).

The slope (m)of a non-vertical line passing through the points (x1,y1) and (x2,y2) is,

m=y2y1x2x1,where x2x1

The slope of the line joining the points (3,1) and (4,2) is,

 m=2(1)43

 =2+1

 =1 

The inclination (θ) of the line joining the points (3,1) and (4,2) is,

tanθ=m

Substitute the value of m.

tanθ=1 

θ=(90+45)

=135

Therefore, the angle between the x-axis and the line joining the points (3,1) and 

(4,2) is 135.


10. The slope of a line is double the slope of another line. If the tangent of the angle between them is 13, find the slope of the lines.

Ans.Take m,m1 as the slopes of the two lines.

Given that, slope of a line is double of the slope of another line.

That is, m1=2m

If θ is the angle between the lines l1 and l2 with slopes m1 and m2 

We have,

tanθ=|m2m11+m1m2|

It is also given that the tangent of the angle between the two lines is 13

13=|m2m1+(2m)m|

13=|m1+2m2|

13=m1+2m2 or

 13=(m1+2m2)=m1+2m2

13=|m1+2m2|

13=m1+2m2 or 

13=(m1+2m2)=m1+2m2

Case 1:

13=m1+2m2

Cross multiply,

1+2m2=3m

2m2+3m+1=0

Rewrite the equation,

2m2+2m+m+1=0

Take out the common terms,

2m(m+1)+1(m+1)=0

(m+1)(2m+1)=0

m=1 or m=12

If m=1, then the slopes of the lines are 1 and 2

Similarly, if m=12, then the slopes of the lines are 12 and 1

Case 2:

13=m1+2m2

Cross multiply,

2m2+1=3m

Equate to zero,

2m23m+1=0

2m22mm+1=0

Take out the common terms,

2m(m1)1(m1)=0

(m1)(2m1)=0

m=1 or m=12

If m=1, then the slopes of the lines are1 and 2

Similarly, if m=12, then the slopes of the lines are 12 and 1 . 

Therefore, the slopes of the lines are 1 and 2 or 12 and 1 or 1 and 2 or 12 and 1 .


11. A line passes through (x1,y1) and (h,k). It slope of the line is m, show that ky1=m(hx1)

Ans.Given that, the slope of the line is m.

The slope of the line passing through (x1,y1) and (h,k) is ky1hx1

Now, ky1hx1=m

ky1=m(hx1)

Therefore, if the line passes through (x1,y1) and (h,k) with a slope of the m

ky1=m(hx1)


Exercise 9.2

1. Write the equation for the x and y-axes.

Ans.The equation of x-axis is y=0 because, y-coordinate of every point on the x-axis 

is zero.

The equation of the y-axis is y=0 because, x-coordinate of every point on the y-axis 

is zero.


2. Find the equation of the line which passes through the point (4,3) with slope 12

Ans.Given that, the point (4,3) and slope of the line is 12.

The equation of the line passing through point (x0,y0), whose slope is m, is 

(yy0)=m(xx0).

Thus, the equation of the line passing through point (4,3), whose slope is 12, is (y3)=12(x+4)

Cross multiply,

2(y3)=x+4

2y6=x+4

Move the constants together,

x2y+10=0

Therefore, the equation of the line which passes through the point (4,3) with slope 12 is x2y+10=0.


3. Find the equation of the line which passes through (0,0) with slope m.

Ans.Given that, the point is  (0,0) and slope is m.

The equation of the line passing through point (x0,y0), whose slope is m, is 

(yy0)=m(xx0).

Substitute the values in the equation,

(y0)=m(x0)

y=mx

Therefore, the equation of the line which passes through (0,0) with slope mis y=mx.


4. Find the equation of the line which passes through (2,23) and is inclined with the x-axis at an angle of 75

Ans.Given that, the point is  (2,23).

The slope of the line that inclines with the x-axis at an angle of 75.

 That is, m=tan75 

m=tan(45+30)

=tan45+tan301tan45tan30

Substitute the values,

=1+131113

 =3+13313

 =3+131 

The equation of the line passing through point (x0,y0), whose slope is m, is (yy0)=m(xx0).

Thus, if a line passes through (2,23) and inclines with the x-axis at an angle of 75,then the equation of the line is,

(y23)=3+131(x2)

(y23)(31)=(3+1)(x2)

y(31)23(31)=x(3+1)2(3+1)

(3+1)x(31)y=23+26+23

(3+1)x(31)y=434

That is, (3+1)x(31)y=4(31)

Therefore, the equation of the line which passes through (2,23) and is inclined with the x-axis at an angle of 75is (3+1)x(31)y=4(31).


5. Find the equation of the line which intersects the x-axis at a distance of 3 units to the left of origin with slope 2.

Ans.Given that, 

Slope, m=2

If a line with slope m makes x-intercept d, then equation of the line is

y=m(xd)

Also given that, the distance is 3units to the left of origin 

That is, d=3

Substitute the values in the equation of line,

y=(2)(x(3))

y=(2)(x+3)

Expand bracket,

y=2x6

2x+y+6=0

Therefore, The equation of the line which intersects the x-axis at a distance of 3 units to the left of origin with slope 2 is 2x+y+6=0.


6. Find the equation of the line which intersects the y-axis at a distance of 2 units above the origin and makes an angle of 30 with the positive direction of the x-axis.

Ans.Given, the line intersects the y-axis at a distance of 2 units above the Origin.

The line makes an angle of 30 with the positive direction of the x-axis.

That is, c=2

m=tan30

=13

If a line with slope m makes y - intercept c, then the equation of the line is,

y=mx+c

Substitute the values,

y=13x+2

y=x+233

3y=x+23

x3y+23=0

Therefore, the equation of the line which intersects the y-axis at a distance of 2 units above the origin and makes an angle of 30 with the positive direction of the x-axis is x3y+23=0.


7.  Find the equation of the line passing through the points (-1,1) and (2,-4)

Ans: We know that the equation passing through the points (x1,y1) and (x2,y2) is yy1=y2y1x2x1(xx1)

The equation of the line that is passing through the points (-1, 1) and (2, -4) is

(y1)=412+1(x(1))

(y1)=53(x+1)

3(y1)=5(x+1)

3y3=5x5

The equation of line is 5x+3y+2=0


8.The vertices of ΔPQR are P(2,1),Q(2,3) and R(4,5). Find equation of the median through the vertex R.

Ans.Given that, the vertices of ΔPQR are P(2,1),Q(2,3) and R(4,5)

Let RL be the median through vertex R.

And L be the mid-point of PQ

By mid-point formula, the coordinates of point L are,

(222,1+32)=(0,2)

The equation of the line passing through points (x1,y1)=(4,5) and (x2,y2)=(0,2) is,

y5=2504(x4)

y5=34(x4)

Cross multiply,

4(y5)=3(x4)

Expand brackets and equate to zero,

4y203x12=0

Rewrite the equation,

3x4y+8=0

Therefore, equation of the median through the vertex Ris 3x4y+8=0.


9. Find the equation of the line passing through (3,5) and perpendicular to the line through the points (2,5) and (3,6).

Ans.The slope of the line joining the points (2,5) and (3,6) is,

m=6532

=15 

It is known that two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other. 

Thus, slope of the line perpendicular to the line through the points (2,5) and (3,6) is,

1m=1(15)=5

The equation of the line passing through point (3,5), whose slope is5is,

 (y5)=5(x+3)

Expand brackets,

y5=5x+15

That is, 5xy+20=0

Therefore, the equation of the line passing through (3,5) and perpendicular to the line through the points (2,5) and (3,6) is 5xy+20=0.


10. A line perpendicular to the line segment joining the points (1,0) and (2,3) divides it in the ratio 1:n. Find the equation of the line.

Ans.By section formula, the coordinates of the point that divides the line segment joining the points (1,0) and (2,3) in the ration 1:n is,

(n(1)+1(2)1+n,n(0)+1(3)1+n)=(n+2n+1,3n+1)

Now, the slope of the line joining the points (1,0) and (2,3) is,

m=3021

=3

It is known that two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other. 

Thus, slope of the line that is perpendicular to the line joining the points (1,0) and (2,3) is,

1m=13

The equation of the line passing through (n+2n+1,3n+1) and whose slope is 13 is ,

(y3n+1)=13(xn+2n+1)

Cross multiply,

3(y3n+1)=(xn+2n+1)

Take LCM,

3[(n+1)y3]=[x(n+1)(n+2)]

3(n+1)y9=(n+1)x+n+2

(1+n)x+3(1+n)y=n+11

Therefore, equation of the line perpendicular to the line segment joining the points (1,0) and (2,3) divides it in the ration 1:n is (1+n)x+3(1+n)y=n+11.


11. Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the points (2,3).

Ans.The equation of a line in the intercept form is xa+yb=1

Here,  a andbare the intercepts on x and y axes respectively. 

Given that,  the line cuts off equal intercepts on both the axes. 

That is, a=b

Now, xa+ya=1

x+y=a

The given line passes through the point (2,3), this equation reduces to

 2+3=a

 a=5 

Substitute the value of a in x+y=a

That is, x+y=5

Therefore, the equation of a line that cuts off equal intercepts on the coordinate axes 

and passes through the points (2,3) is x+y=5.


12. Find the equation of the line passing through the points (2,2) and cutting off intercepts on the axes whose sum is 9 .

Ans.The equation of the line making intercepts a and b on x-and y-axis respectively is,

xa+yb=1..(1)

Given that, sum of intercepts =9

That is, a+b=9

b=9a

Substitute value of b in the above equation, 

xa+y(9a)=1

Also given that, the line passes through the point (2,2)

Then, 2a+2(9a)=1

2(9a)+2aa(9a)=1

182a+2aa(9a)=1

18a(9a)=1

18=a(9a)

18=9aa2

a29a+18=0

Factorize the equation, 

a23a6a+18=0

a(a3)6(a3)=0

(a3)(a6)=0

a=3 or a=6

Substitute in (1),

Case 1:(a=3) 

b=93=6

x3+y6=1

2x+y=6

Equate to zero,

2x+y6=0

Case 2:(a=6) 

b=96=3

x6+y3=1

x+2y=6

Equate to zero,

x+2y6=0

Therefore, the equation of the line is 2x+y6=0 or x+2y6=0.


13. Find equation of the line through the points (0,2) making an angle 2π3 with the positive x axis. Also, find the equation of the line parallel to it and crossing the y-axis at a distance of 2 units below the origin.

Ans.The slope of the line making an angle 2π3 with the positive x-axis is,

m=tan(2π3)

=3

The equation of the line passing through points (0,2) and having a slope 3 is,

(y2)=3(x0)

That is, 3x+y2=0

The slope of line parallel to line 3x+y2=0 is 3

Given that the line parallel to line 3x+y2=0 crosses the y-axis 2 units below the origin.

It passes through point (0,2)

Thus, the equation of the line passing through points (0,2) and having a slope 3 is,

y(2)=3(x0)

y+2=3x

3x+y+2=0

Therefore, the equation of the line through the points (0,2) making an angle 2π3 with the positive x axis is 3x+y2=0 and the equation of the line parallel to it and crossing the y-axis at a distance of 2 units is 3x+y+2=0.


14. The perpendicular from the origin to a line meets it at the point (2,9), find the equation of the line.

Ans.Given that, The perpendicular from the origin to a line meets it at the point (2,9).

The slope of the line joining the origin (0,0) and point (2,9) is,

m1=9020

=92

Then, the slope of the line perpendicular to the line joining the origin and points (2,9) is,

m2=1m1

=1(92)

=29

Now, the equation of the line passing through point (2,9) and having a slope m2 is,(y9)=29(x+2)

Cross multiply and expand brackets,

9y81=2x+4

That is, 2x9y+85=0

Therefore, the equation of the line is 2x9y+85=0.


15. The length L (in centimeter) of a copper rod is a linear function of its Celsius temperature C. In an experiment, if L=124.942 when C=20 and L=125.134 when 

C=110, express L in terms of C.

Ans.It is given that when C=20L=124.942 and when C=110L=125.134

The points (20,124.942) and (110,125.134) satisfy the linear relation between Land C.

Assume Calong the x-axis and L along the y-axis, we have two points, (20,124.942) 

and (110,125.134) in the XY plane. 

Thus, the linear relation between L and C is the equation of the line passing through 

thepoints (20,124.342) and (110,125.134).

Now, (L124.942)=125.134124.94211020(C20)

(L124.942)=0.19290(C20)

L=0.19290(C20)+124.942.


16. The owner of a milk store finds that, he can sell 980 liters of milk each week at Rs 14/ liter and1220 liters of milk each week at Rs 16/ liter. Assuming a linear relationship between selling price and demand, how many liters could he sell weekly at Rs 17/ liter?

Ans.Given that, the owner can sell 980 liters of milk each week at Rs 14/ liter and 1220 liters of milk each week at Rs 16/ liter. 

The relationship between the selling price and demand is linear. 

Assume selling price per liter along the x-axis and demand along the y-axis, we have two points (14,980) and (16,1220) in the XY plane that satisfy the linear relationship between selling price and demand.

Thus, the line passing through points (14,980) and (16,1220)

That is, y980=12209801614(x14)

y980=2402(x14)

y980=120(x14)

y=120(x14)+980

If x= Rs 17/ liter, 

y=120(1714)+980

y=120×3+980

 =360+980

 =1340

Therefore, the owner of the milk store could sell 1340 liters of milk weekly at Rs 17/liter.


17. P(a,b) is the mid-point of a line segment between axes. Show that the equation of the line is xa+yb=2

Ans.Let AB be a line segment whose midpoint is P(a,b).

Let the coordinates of A and B be (0,y) and (x,0) respectively.


a line segment whose midpoint


The mid-pint is,

(0+x2,y+02)=(a,b)

(x2,y2)=(a,b)

x2=a and y2=b

x=2a and y=2b

Now, the respective coordinates of A and B are (0,2b) and (2a,0).

The equation of the line passing through points (0,2b) and (2a,0) is,

(y2b)=(02b)(2a0)(x0)

y2b=2b2a(x)

Cancel 2and cross multiply,

a(y2b)=bx

ay2ab=bx

That is, bx+ay=2ab

Divide both sides by ab

bxab+ayab=2abab

x2+yh=2

Hence, if P(a,b) is the mid-point of a line segment between axes then equation of the line is xa+yb=2.


18.Point R(h,k) divides a line segment between the axes in the ratio1:2. Find the equation of the line.

Ans.Consider, ABbe the line segment such that r(h,k) divides it in the ratio 1:2.

Then, the coordinates of A and B be (0,y) and (x,0) respectively.


the coordinates of A and B be (0,y) and (x,0) respectively


By section formula,

(h,k)=(1×0+2×x1+2,1×y+2×01+2)

(h,k)=(2x3,y3)

h=2x3 and k=y3

x=3h2 and y=3k

Thus, the respective coordinates of A and B are (3h2,0) and (0,3k).

Now, the equation of the line AB passing through points (3h2,0) and (0,3k) is,(y0)=3k003h2(x3h2)

y=2kh(x3h2)

hy=2kx+3hk

Rewrite the equation,

2kx+hy=3hk

Therefore, the equation of the line is 2kx+hy=3hk.


19.By using the concept of equation of a line, prove that the three points (3,0),(2,2) and (8,2) are collinear.

Ans.It is necessary to show that the line passing through points (3,0) and (2,2) also passes through point (8,2)in order to show that the points (3,0),(2,2) and (8,2) are collinear.

The equation of the line passing through points (3,0) and (2,2) is,

(y0)=(20)(23)(x3)

y=25(x3)

Cross multiply and expand bracket,

5y=2x6

Rewrite the equation,

2x5y=6

At x=8 and y=2,

2×85×2=1610

=6

Thus, the line passing through points (3,0) and (2,2) also passes through point (8,2).

Therefore, the points (3,0),(2,2), and (8,2) are collinear.


Exercise 9.3

1.Reduce the following equation into slope-intercept form and find their slopes and the y-intercepts.

(i) x+7y=0

Ans.Given that, the equation is x+7y=0

Slope - intercept form is represented as y=mx+c, where m is the slope and c is the y Intercept.

Now, the equation can be expressed as,

y=17x+0

Therefore, the above equation is of the form y=mx+c, where m=17 and c=0.

(ii) 6x+3y5=0

Ans.Given that, the equation is 6x+3y5=0

Slope - intercept form is represented as y=mx+c, where m is the slope and c is the yintercept.

Now, the equation can be expressed as,

3y=6x+5

y=63x+53

=2x+53

Therefore, the above equation is of the form y=mx+c, where m=2 and c=53.

(iii) y=0

Ans.Given that, the equation is y=0

Slope - intercept form is given by y=mx+c, where m is the slope and c is the y intercept.

Then, y=0×x+0

Therefore, the above equation is of the form y=mx+c, where m=0 and c=0.


2.Reduce the following equations into intercept form and find their intercepts on the axes.

(i) 3x+2y12=0

Ans.Given that, the equation is 3x+2y12=0

Rewrite the equation,

3x+2y=12

Divide both sides by 12,

3x12+2y12=1

x4+y6=1

This equation is of the form xa+yb=1, where a=4 and b=6

Therefore, the equation is in the intercept form, where the intercepts on the x and y axes are 4and 6respectively.

(ii) 4x3y=6

Ans.Given that, the equation is 4x3y=6

Divide both sides by 6,

4x63y6=1

2x3y2=1

x(32)+y(2)=1

Therefore, the equation is in the intercept form, where the intercepts on x and yaxes are 32 and 2 respectively.

(iii) 3y+2=0

Ans.Given that, the equation is 3y+2=0

Rewrite the equation,

3y=2

Divide both sides by 2,

y(23)=1

Now, equation is in the form xa+yb=1, where a=0 and b=23

Therefore, the equation is in the intercept form, where the intercept on the y-axis is 23 and it has no intercept on the x-axis.


3. Find the distance of the points (1,1) from the line 12(x+6)=5(y2).

Ans.Given that, the equation of the line is 12(x+6)=5(y2)

Expand brackets,

12x+72=5y10

Rewrite the equation,

12x5y+82=0

When comparing this equation with general equation of line Ax+By+C=0, we get

A=12, B=5, and C=82

We know that the perpendicular distance (d) of a lineAx+By+C=0 from a point 

(x1,y1) is, 

d=|Ax1+By1+C|A2+B2

The given point is (x1,y1)=(1,1)

Thus, the distance of point (1,1) from the given line is,

|12(1)+(5)(1)+82|(12)2+(5)2 units 

=|125+82|169 units 

=|65|13 units 

=5 units

Therefore, the distance of the points (1,1) from the line 12(x+6)=5(y2) is 5 units.


4. Find the points on the x-axis whose distance from the line x3+y4=1 are 4 units.

Ans.Given that, the equation of line is x3+y4=1

It can be write as 4x+3y12=0

When comparing this equation with general equation of line Ax+By+C=0, we get

A=4,B =3, and C=12

Let (a,0) be the point on the x-axis whose distance from the given line is 4 units. 

We know that the perpendicular distance (d) of a line Ax+By+C=0 from a point 

(x1,y1) is d=|Ax1+By1+C|A2+B2

Thus, 4=|4a+3×012|42+32

4=|4a12|5

|4a12|=20

±(4a12)=20

(4a12)=20 or (4a12)=20

4a=20+12 or 4a=20+12

a=8 or 2

Therefore, the required points on x-axis are (2,0) and (8,0).


5. Find the distance between parallel lines

(i) 15x+8y34=0 and 15x+8y+31=0

Ans.We know that the distance(d) between parallel lines Ax+By+C1=0 and 

Ax+By+C2=0 is,

 d=|C1C2|A2+B2

The given parallel lines are 15x+8y34=0 and 15x+8y+31=0 

Here, A=15,B=8,C1=34, and C2=31.

Thus, the distance between the parallel lines is,

d=|C1C2|A2+B2=|3431|(15)2+(8)2 units 

=|65|289 units 

=6517 units

Therefore, the distance between parallel lines 15x+8y34=0 and 15x+8y+31=0 is 

6517 units.

(ii) l(x+y)+p=0 and l(x+y)r=0

Ans.The given parallel lines are l(x+y)+p=0 and l(x+y)r=0

It can be write as,

lx+ly+p=0 and lx+lyr=0

Here, A=l,B=l,C1=p, and C2=r.

Thus, the distance between the parallel lines is,

d=|C1C2|A2+B2=|p+r|l2+l2 units 

=|p+r|2l2 units 

=|p+r|l2 units 

=12|p+r|l units

Therefore, the distance between parallel lines l(x+y)+p=0 and l(x+y)r=0

is 12|p+r|l units.


6. Find equation of the line parallel to the line 3x4y+2=0 and passing through the point (2,3).

Ans.Here, the equation of the given line is 3x4y+2=0

Rewrite as,

y=3x4+24

This can be write as,

y=34x+12 , which is of the form y=mx+c

Now, slope of the given line =34 

It is known that parallel lines have the same slope.

Thus, slope of the other line is m=34 

Now, the equation of the line which has a slope of 34 and passes through the points 

(2,3) is,

(y3)=34{x(2)}

Cross multiply and expand brackets,

4y12=3x+6

Rewrite as,

3x4y+18=0

Therefore, the equation of the line parallel to the line 3x4y+2=0 and passing 

through the point (2,3) is 3x4y+18=0.


7. Find the equation of the line perpendicular to the line x7y+5=0 and having x intercept 3.

Ans.The given equation of the line is x7y+5=0.

It can be write as,

y=17x+57, which is of the form y=mx+c

Now, Slope of the given line =17 

The slope of the line perpendicular to the line having a slope of 17 is,

m=1(17)=7

The equation of the line with slope 7 and x-intercept 3 is,

y=m(xd)

Substitute the values,

y=7(x3)

Expand bracket,

y=7x+21

7x+y=21

Therefore, the equation of the line perpendicular to the line x7y+5=0 and having x

intercept 3 is 7x+y=21.

 

8. Find angles between the lines 3x+y=1 and x+3y=1.

Ans.The given lines are 3x+y=1 and x+3y=1 

Consider the first line,

3x+y=1 can be write as,

y=3x+1 

The slope of this line is m1=3.

Now, consider the second line,

x+3y=1 can be write as,

y=13x+13

The slope of this line is m2=13

The acute angle that is, θ between the two lines is,

tanθ=|m1m21+m1m2|

Substitute the values,

tanθ=|3+131+(3)(13)|

tanθ=|3+131+1|

 tanθ=|22×3|

tanθ=13

θ=30

Thus, the angle between the lines 3x+y=1 and x+3y=1 is either 30 or 

18030=150.


9. The line through the points (h,3) and (4,1) intersects the line 7x9y19=0At right angle. Find the value of h.

Ans.The slope of the line passing through points (h,3) and (4,1) is,

m1=134h

=24h

Given that, the line 7x9y19=0  

It can be write as,

y=79x199 

The slope off this line is,

m2=79.

It is given that the two lines are perpendicular. 

Then,  m1×m2=1

Substitute the values,

14369h=1

Cross multiply,

14=369h

Rewrite as,

9h=3614

h=229

Therefore, the value of h is 229.


10. Prove that the line through the point (x1,y1) and parallel to the line Ax+By+C=0 is A(xx1)+B(yy1)=0

Ans.One of the line is Ax+By+C=0  

Then, y=(AB)x+(CB) 

The slope of this line is,

 m=AB

We know that parallel lines have the same slope.

Hence, Slope of the other line is m=AB

The equation of the line passing through point (x1y1) and having a slope  m=AB

is,

yy1=m(xx1)

Substitute the value of m,

yy1=AB(xx1)

Cross multiply,

B(yy1)=A(xx1)

Rewrite as,

A(xx1)+B(yy1)=0

Therefore, the line through point (x1y1) and parallel to line Ax+By+C=0 is 

A(xx1)+B(yy1)=0.


11. Two lines passing through the points (2,3) intersects each other at an angle of 60. If slope of one line is 2 , find equation of the other line.

Ans.Given that the slope of the first line is 2

That is, m1=2

Assume that the slope of the other line is m2

The angle between the two lines is 60.

We have, tan60=|m1m21+m2m2|

Substitute the known values,

3=|2m21+2m2|

3=±(2m21+2m2)

3=2m21+2m2  or  3=(2m21+2m2)

3(1+2m2)=2m2  or  3(1+2m2)=(2m2)

3+23m2+m2=2  or  3+23m2m2=2

3+(23+1)m2=2  or  3+(231)m2=2

m2=23(23+1)  or  m2=(2+3)(231)

Case 1:

m2=(23(23+1))

The equation of the line passing through point (2,3) and having a slope of 23(23+1) is,

(y3)=(23)(23+1)(x2)

(23+1)y3(23+1)=(23)x(23)2

(32)x+(23+1)y=4+23+63+3

(32)x+(23+1)y=1+83

Now, the equation of the other line is (32)y+(23+1)y=1+83.

Case 2:

m2=(2+3)(231)

The equation of the line passing through points (2,3) and having a slope of (2+3)(231) is,

(y3)=(2+3)(231)(x2)

(231)y3(231)=(231)x+2(231)

(231)y+(231)x=4+23+633

(2+3)x+(231)y=1+83

Now, the equation of the other line is (2+3)x+(231)y=1+83.

Therefore, the required equation of the other line is (32)x+(23+1)y=1+83 or 

(2+3)x+(231)y=1+83.


12. Find the equation of the right bisector of the line segment joining the points (3,4) and (1,2).

Ans.The right bisector of a line segment bisects the line segment at an angle 90

The end-points of the line segment are given as A(3,4) and B(1,2)

Now, mid-point of AB=(312,4+20)=(1,3)

Slope of AB==2413=24=12

Thus, Slope of the line perpendicular to AB=1(12)=2 

The equation of the line passing through (1,3) and having a slope of 2 is,

(y3)=2(x1)

Expand bracket,

y3=2x+2

2x+y=5

Therefore, the equation of the right bisector of the line segment joining the points (3,4) 

and (1,2) is 2x+y=5.


13. Find the coordinates of the foot of perpendicular from the points (1,3) to the line 3x4y16=0.

Ans.Let (a,b) be the coordinates of the foot of the perpendicular from the points (1,3) 

to the line3x4y160.

Slope of the line joining (1,3) and (a,b) is,

m1=b3a+1

3x4y16=0 can be write as,

y=34x4

Slope of the line is,

m2=34

Since these two lines are perpendicular, 

m1m2=1 (b3a+1)×(34)=1

3b94a+4=1

3b9=4a4

4a+3b=5(1)

Point (a,b) lies on line 3x4y=16.

Rewrite as,

3a4b=16(2)

Solving equations (1) and (2), 

We get a=6825 and b=4925

Therefore, the required coordinates of the foot of the perpendicular are (6825,4925).


14. The perpendicular from the origin to the line y=mx+c meets it at the point (1,2). Find the values of m and c.

Ans.The given equation of line is y=mx+c

It is also given that the perpendicular from the origin meets the given line at (1,2)

Then, the line joining the points (0,0) and (1,2) is perpendicular to the given line.

Now, slope of the line joining (0,0) and (1,2)=21=2 

The slope of the given line is m.

 m×2=1 (The two lines are perpendicular )

m=12

Since points (1,2) lies on the given line, it satisfies the equation y=mx+c.

2=m(1)+c

2=2+12(1)+c

c=2+12=52

Therefore, the values of m and c are 12 and 52 respectively.


15. If p and q are the lengths of perpendicular from the origin to the lines xcosθysinθ=k cos2θ and xsecθ+ycosecθ=k, respectively, prove that  p2+4q2=k2

Ans.The equation of given lines are,

xcosθysinθ=kcos2θ(1) 

xsecθ+ycosecθ=k(2)

The perpendicular distance (d)of a line Ax+By+C=0 from a point (x1,x2) is,d=|Ax1+By1+C|A2+B2

Compare equation (1) to the general equation of line that is., Ax+By+C=0

We get  A=cosθ,B=sinθ, and C=kcos2θ

It is given that p is the length of the perpendicular from (0,0) to line (1).

p=|A(0)+B(0)+C|A2+B2

=|C|A2+B2

=|kcos2θ|cos2θ+sin2θ

=|kcos2θ|(3)

Compare equation (2)to the general equation of line that is, Ax+By+C=0

We get A=secθ,B=cosecθ, and C=k.

It is given that q is the length of the perpendicular from (0,0) to line (2).

q=|A(0)+B(0)+C|A2+B2

=|C|A2+B2

=|k|sec2θ+cosec2θ(4)

From (3)and(4), we have

p2+4q2=(|kcos2θ|)2+4(|k|sec2θ+cosec2θ)2

=k2cos22θ+4k2(sec2θ+cosec2θ)

=k2cos22θ+4k2(1cos2θ+1sin2θ)

=k2cos22θ+4k2(sin2θ+cos2θsin2θcos2θ)

=k2cos22θ+4k2(1sin2θcos2θ)

=k2cos22θ+4k2sin2θcos2θ

=k2cos22θ+k2(2sinθcosθ)2

=k2cos22θ+k2sin22θ

=k2(cos22θ+sin22θ)

=k2

Hence, it is proved that p2+4q2=k2.


16. In the triangle ABC with vertices A(2,3),B(4,1) and C(1,2), find the equation and length of altitude from the vertex A.

Ans.Let ADbe the altitude of triangle ABCfrom vertex A

So, ADBC


the altitude of triangle ABC from vertex A


The equation of the line passing through point (2,3) and having a slope of 1 is,

(y3)=1(x2)

xy+1=0

yx=1

Thus, equation of the altitude from vertex A is yx=1.

Length of AD= Length of the perpendicular from A(2,3) to BC 

The equation of BC is,

(y+1)=2+114(x4)

(y+1)=1(x4)

Open brackets,

y+1=x+4

x+y3=0(1)

The perpendicular distance (d) of a line Ax+By+C=0 from a point (x1,y1) is,

 d=|Ax1+By1+C|A2+B2

Compare equation (1) to the general equation of line Ax+By+C=0

We get, A=1, B=1, and C=3.

Length of AD=|1×2+1×33|12+12 units 

=|2|2 units 

=22 units 

=2 units

Therefore, the equation and length of the altitude from vertex A are yx=1 and 2 

units respectively.


17. If p is the length of perpendicular from the origin to the line whose intercepts on the axes area, and b, then show that 1p2=1a2+1b2.

Ans.The equation of a line whose intercepts on the axes are a and b is,

 xa+yb=1

Or bx+ay=ab

Or bx+ayab=0(1)

The perpendicular distance (d)of a line Ax+By+C=0 from a point (x1,y1) is,

d=|Ax1+By1+C|A2+B2

Compare equation (1)to the general equation of line AxBy+C=0

We get, A=b, B=a, and C=ab.

Thus, if p is the length of the perpendicular from point (x1,y1)=(0,0) to line(1)

We get, p=|A(0)+B(0)ab|b2+a2

p=|ab|b2+a2

Square both sides, 

p2=(ab)2a2+b2

p2(a2+b2)=a2b2

a2+b2a2b2=1p2

1p2=1a2+1b2

Hence, it is shown that 1p2=1a2+1b2.


Miscellaneous Exercise

1. Find the value of k for which the line (k3)x(4k2)y+k27k+6=0 is

(a) Parallel to x-axis

Ans.The given equation of line is (k3)x(4k2)y+k27k+6=0

If the line is parallel to x-axis,

Slope of the line = Slope of the x-axis

It can be written as,

(4k2)y=(k3)x+k27k+6=0

We get,

y=(k3)(4k2)x+k27k+6(4k2), Which is of the form y=mx+c

Here, the slope of the given line=(k3)(4k2)

Consider the slope of x-axis =0

(k3)(4k2)=0

k3=0

k=3

Therefore, if the given line is parallel to the x-axis, then the value of k is 3 .

(b) Parallel to y-axis

Ans.The given equation of line is (k3)x(4k2)y+k27k+6=0

Here if the line is parallel to the y-axis, it is vertical and the slope will be undefined.

So, the slope of the given line=(k3)(4k2)

Here, (k3)(4k2) is undefined at k2=4

k2=4

k=±2

Therefore, if the given line is parallel to the y-axis, then the value of k is ±2.

(c) Passing through the origin.

Ans.The given equation of line is (k3)x(4k2)y+k27k+6=0

Here, if the line is passing through (0,0) which is the origin satisfies the given equation 

of line,

(k3)(0)(4k2)(0)+k27k+6=0

k27k+6=0

Separate the terms,

k26kk+6=0

(k6)(k1)=0

k=1or 6 

Therefore, if the given line is passing through the origin, then the value of k is either 

1or 6.


2. Find the equation of the line, which cut-off intercepts on the axes whose sum and product are 1and 6, respectively.

Ans.Consider, the intercepts cut by the given lines on the axes are aand b.

a+b=1(1) (1)

ab=6(2)

Solve both the equations to get

a=3 and b=2 or a=2 and b=3

We know that the equation of the line whose intercepts on a and b axes is

xa+yb=1 or bx+ayab=0

Case 1: a=3 and b=2

Now, the equation of the line is 2x+3y+6=0

That is, 2x3y=6

Case 2: a=2 and b=3

Now, the equation of the line is 3x2y+6=0

That is,3x+2y=6

Therefore, the required equation of the lines are 2x3y=6 and 3x+2y=6.


3. What are the points on the y-axis whose distance from line x3+y4=1 is 4 units.

Ans.Consider (0,b) as the point on the y-axis whose distance from line x3+y4=1 is  4 units.

It can be written as 4x+3y12=0(1)

Compare equation (1) to the general equation of line Ax+Bx+C=0, we get

A=4,B=3 and C=12

We know that the perpendicular distance (d) of a line Ax+By+C=0 from (x1,y1) is,

d=|Ax1+By1+C|A2+B2

If (0,b) is the point on the y-axis whose distance from line x3+y4=1  is 4 units, then

4=|4(0)+3(b)12|42+32

4=|3b12|5

By cross multiplication,

20=|3b12|

20=±(3b12)

Here, 20=(3b12) or 20=(3b12)

It can be written as

3b=20+12 or 3b=20+12

Now we get,

b=323 or b=83

Therefore, the required points are b=323 and b=83.


4. Find the perpendicular distance from the origin to the line joining the points (cosθ,sinθ)and(cosϕ,sinϕ)

Ans.The equation of the line joining the points (cosθ,sinθ) and (cosϕ,sinϕ) is,

(cosθ,sinθ)and(cosϕ,sinϕ)

ysinθ=sinϕsinθcosϕcosθ(xcosθ)

By cross multiplication,

y(cosϕcosθ)sinθ(cosϕcosθ)=x(sinϕsinθ)cosθ(sinϕsinθ)

x(sinθsinϕ)+y(cosϕcosθ)+cosθsinϕcosθsinθsinθcosϕ+sinθcosθ=0

x(sinθsinϕ)+y(cosϕcosθ)+sin(ϕθ)=0

Ax+By+C=0, where A=sinθsinϕ,B=cosϕcosθ, and C=sin(ϕθ)

It is known that the perpendicular distance (d)of a line Ax+By+C=0 from a point 

(x1,y1) is,

d=|Ax1+By1+C|A2+B2

Thus, the perpendicular distance (d) of the given line from point (x1,y1)=(0,0) is,

d=|(sinθsinϕ)(0)+(cosϕcosθ)(0)+sin(ϕθ)|(sinθsinϕ)2+(cosϕcosθ)2

=|sin(ϕθ)|sin2θ+sin2ϕ2sinθsinϕ+cos2ϕ+cos2θ2cosϕcosθ

Group the terms,

=|sin(ϕθ)|(sin2θ+cos2θ)+(sin2ϕ+cos2ϕ)2(sinθsinϕ+cosϕcosθ)

=|sin(ϕθ)|1+12(cos(ϕθ))

=|sin(ϕθ)|2(1cos(ϕθ))

=|sin(ϕθ)|2(2sin2(ϕθ2))

=|sin(ϕθ)|2sin(ϕθ2)

Therefore, the perpendicular distance from the origin to the line joining the points 

(cosθ,sinθ)and(cosϕ,sinϕ)is |sin(ϕθ)|2sin(ϕθ2).


5.Find the equation of the line parallel to y-axis and draw through the point of intersection of the lines x7y+5=0 and 3x+y=0.

Ans.The equation of any line parallel to the y-axis is of the form x=a(1)

The two given lines are x7y+5=0(2)

3x+y=0(3)

Solve equation (2) and (3), we get x=522 and y=1522 

Thus, (522,1522) is the point of intersection of lines (2) and (3).

Since line x=a passes through point (522,1522),

a=522

Therefore, the required equation of the line is x=522.


6.Find the equation of a line drawn perpendicular to the line x4+y6=1 through the point, where it meets the y-axis.

Ans.Here, the equation of the given line is x4+y6=1 

This equation can be written as 3x+2y12=0 

Rewrite as,

y=32x+6, which is of the form y=mx+c

Now, Slope of the given line =32

Slope of line perpendicular to the given line =1(32)=23 

Let the given line intersect the y-axis at (0,y)

Substitute x with 0 in the equation of the given line, 

y6=1y=6 

The given line intersects the y-axis at (0,6)

The equation of the line that has a slope of 23 and passes through point (0,6) is,(y6)=23(x0)

Cross multiply and expand brackets,

3y18=2x

2x3y+18=0

Therefore, the required equation of the line is 2x3y+18=0.


7.Find the area of the triangle formed by the line yx=0,x+y=0 and xk=0.

Ans.It is given that,

yx=0(1) 

x+y=0(2)

xk=0(3)

Here, the point of intersection of lines (1)and (2) is,

x=0 and y=0

The point of intersection of lines (2) and (3) is,

x=k and y=k

The point of intersection of lines (3)and (1) is,

x=k and y=k

Now, the vertices of the triangle formed by the three given lines are (0,0),(k,k) and 

(k,k).

Here the area of triangle whose vertices are (x1,y1),(x2,y2) and (x3,y3) is,

12|x1(y2y3)+x2(y3y1)+x3(y1y2)|

So the area of triangle formed by the three given lines,

=12|0(kk)+k(k0)+k(0+k)| square units

=12|k2+k2|square units

We get,

=12|2k2|

=k2 square units

Therefore, the area of the triangle formed by the line yx=0,x+y=0 and xk=0 is 

k2 square units.


8.Find the value of p so that the three lines 3x+y2=0,px+2y3=0 and 2xy3=0 may intersect at one point.

Ans.It is given that,

3x+y2=0(1)

px+2y3=0(2)

2xy3=0(3)

Solve equations (1) and (3) to get

x=1 and y=1

Here, the three lines intersect at one point and the point of intersection of lines (1) and

(3)will also satisfy line (2)

p(1)+2(1)3=0

p23=0

p=5

Therefore, the value of p is 5.


9.If three lines whose equations are y=m1x+c1,y=m2x+c2, and y=m2x+c2 are concurrent, then show that m1(c2c3)+m2(c3c1)+m3(c1c2)=0.

Ans.It is given that,

y=m1x+c1(1)

y=m2x+c2(2)

y=m3x+c3(3)

Subtract equation (1)from (2),

0=(m2m1)x+(c2c1)

(m1m2)x=c2c1

x=c2c1m1m2

Substitute this value in equation (1) 

y=m1(c2c1m1m2)+c1

Multiply the terms,

y=m1c2m1c1m1m2+c1

Take LCM,

y=m1c2m1c1+m1c1m2c1m1m2

y=m1c2m2c1m1m2

Here,

(c2c1m1m2,m1c2m2c1m1m2) is the point of intersection of lines (1) and (2)

Lines (1), (2)and (3) are concurrent. So the point of intersection of lines (1) and (2)

 will satisfy equation (3).

m1c2m2c1m1m2=m3(c2c1m1m2)+c3

Multiply the terms and take LCM,

m1c2m2c1m1m2=m3c2m3c1+c3m1c3m2m1m2

By cross multiplication,

m1c2m2c1m3c2+m3c1c3m1+c3m2=0

Take out the common terms,

m1(c2c3)+m2(c3c1)+m3(c1c2)=0

Therefore, If three lines whose equations are y=m1x+c1,y=m2x+c2, and y=m2x+c2 

are concurrent, then m1(c2c3)+m2(c3c1)+m3(c1c2)=0.


10.Find the equation of the line through the points (3,2) which make an angle of 45 with the line x2y=3

Ans.Let the slope of the required line be m1

The given line can be written as,

 y=12x32, which is of the form y=mx+c

Now, slope of the given line is 

m2=12 

It is given that the angle between the required line and line x2y=3 is 45

If θ is the acute angle between lines l1 and l2 with slopes m1 and m2 respectively, then

tanθ=|m2m11+m1m2|

Now, tan45=|m2m1|1+m1m2

Substitute the values,

1=|12m11+m12|

Take LCM,

1=|(12m12)2+m12|

1=|12m12+m1|

1=±(12m12+m1)

1=12m12+m1 or 1=(12m12+m1)

2+m1=12m1 or 2+m1=1+2m1

m1=13 or m1=3

Case 1: m1=3 

The equation of the line passing through (3,2) and having a slope of 3 is,

y2=3(x3)

Expand bracket,

y2=3x9

3xy=7

Case 2: m1=13

The equation of the line passing through (3,2) and having a slope of 13 is

y2=13(x3)

Cross multiply and expand bracket,

3y6=x+3

x+3y=9

Therefore, the equations of the line are 3xy=7 and x+3y=9.


11.Find the equation of the line passing through the point of intersection of the line 4x+7y3 =0 and 2x3y+1=0 that has equal intercepts on the axes.

Ans.Let the equation of the line having equal intercepts on the axes be xa+ya=1

It can be written as,

x+y=a(1)

Solve equations 4x+7y3=0 and 2x3y+1=0, we get x=113 and y=513

(113,513) is the point of the intersection of the two given lines. 

Since equation (1) passes through point (113,513), 113+513=a

a=613

Thus, Equation (1) becomes x+y=613 that is, 13x+13y=6

Therefore, the required equation of the line 13x+13y=16.


12.Show that the equation of the line passing through the origin and making an angle θ with the line y=mx+c, is yx=m±tanθ1mtanθ

Ans.Let the equation of the line passing through the origin be y=m1x

If this line makes an angle of θ with line y=mx+c, then angle θ is,

tanθ=|m1m1+m1m|

Substitute the values,

tanθ=|yxm1+yxm|

tanθ=±(yxm1+yxm)

tanθ=yxm1+yxm or tanθ=(yxm1+yxm)

Case 1: tanθ=yxm1+yxm

tanθ+yxmtanθ=yxm

m+tanθ=yx(1mtanθ)

yx=m+tanθ1mtanθ

Case 2: tanθ=(xym1+yxm)

tanθ=(yxm1+yxm)

tanθ+yxmtanθ=yx+m

yx(1+mtanθ)=mtanθ

Rewrite as,

yx=mtanθ1+mtanθ

Hence, it is shown that the equation of the line passing through the origin and making an angle θwith the line y=mx+c, is yx=m±tanθ1mtanθ.


13.In what ratio, the line joining (1,1) and (5,7) is divisible by the line x+y=4 ?

Ans.The equation of the line joining the points (1,1) and (5,7) is,

y1=715+1(x+1)

y1=66(x+1)

xy+2=0(1)

The equation of the given line is x+y4=0(2).

The points of intersection of line (1) and (2) is x=1 and y=3.

Let point (1,3) divide the line segment joining (1,1) and (5,7) in the ratio 1:k.

Then, by section formula, 

(1,3)=(k(1)+1(5)1+k,k(1)+1(7)1+k)

Simplify,

(1,3)=(k+51+k,k+71+k)

k+51+k=1,k+71+k=3

k+51+k=1

By cross multiplication,

k+5=1+k

2k=4

k=2
Therefore, the line joining the points (1,1) and (5,7) is divided by line x+y=4 in the ratio 1:2.


14. Find the distance of the line 4x+7y+5=0 from the point (1,2) along the line 2xy=0

Ans.The given lines are 2xy=0(1)

4x+7y+5=0(2)

P(1,2) is a point on line(1)

Let Q be the point intersection of line (1) and (2).


The point intersection of line


Solve equations (1) and (2) to get x=518 and y=59

Now, Coordinates of point B are (518,59)

Use distance formula to obtain the distance between points P and Q,

PQ=(1+518)2+(2+59)2 units

Take LCM,

=(2318)2+(239)2 units

Rewrite as,

=(232×9)2+(239)2 units

=(239)2(12)2+(239)2 units

=(239)2+(14+1) units

=23954 units

=239×52 units

=23518 units

Therefore, the required distance is 23518 units,


15. Find the direction in which a straight line must be drawn through the points (1,2) so that its point of intersection with line x+y=4 may be at a distance of 3 units from this point.

Ans.Consider y=mx+c as the line passing through the point (1,2).

Then, 

2=m(1)+c

2=m+c

c=m+2

Substitute the value of c,

y=mx+m+2(1) 

Now, the given line is,

x+y=4(2)

Solve both equations,

x=2mm+1 and y=5m+2m+1

(2mm+1,5m+2m+1) is the point of intersection of lines (1) and (2).

Given that, the point is at a distance of 3 units from (1,2)

By distance formula,

(2mm+1+1)2+(5m+2m+12)2=3

Square both sides,

(2m+m+1m+1)2+(5m+22m2m+1)2=32

9(m+1)2+9m2(m+1)2=9

Divide the equation by 9,

1+m2(m+1)2=1

By cross multiplication,

1+m2=m2+1+2m

2m=0

m=0

Therefore, the slope of the required line must be zero that is, the line must be parallel 

to the x-axis.


16. The hypotenuse of a right-angled triangle has its ends at the points (1,3) and (4,1). Find the equation of the legs (perpendicular sides) of the triangle.

Ans.Consider PQR as the right angles triangle where R=90

Here, infinity such lines are present.

m is the slope of PR

Then, the slope of QR=1m

Equation of PRis,

y3=m(x1)

By cross multiplication,

x1=1m(y3)

Equation of QRis,

y1=1m(x+4)

By cross multiplication

x+4=m(y1)

If m=0,

y3=0,x+4=0

If m=,

x1=0,y1=0 

That is, x=1,y=1

Therefore, the equation of the legs (perpendicular sides) of the triangle is x=1,y=1.


17. Find the image of the point (3,8) with respect to the line x+3y=7 assuming the line to be a plane mirror.

Ans.Given that,

x+3y=7(1)

Consider B(a, b)as the image of point A(3,8)

So line (1)is perpendicular bisector of AB.


perpendicular bisector of


Slope of AB=b8a3

Slope of line (1)=13

Line (1) is perpendicular to AB

Then,

(b8a3)×(13)=1

b83a9=1

By cross multiplication,

b8=3a9

3ab=1(2)

We know,

Mid-point of AB=(a+32,b+82)

So the mid-point of line segment AB will satisfy line (1).

From equation (1),

(a+32)+3(b+82)=7

By further calculation,

a+3+3b+24=14

On further simplification,

a+3b=13(3)

Solve equations (2) and (3),

a=1 and b=4

Therefore, the image of the given point with respect to the given line is (1,4).


19.If the lines y=3x+1 and 2y=x+3 are equally indicated to the line y=mx+4, find the value of m.

Ans.The equation of the given lines are y=3x+1

2y=x+3(2)

y=mx+4(3)

Slope of line(1) is  m1=3

Slope of line (2)ism2=12

Slope of line (3)ism3=m

We know that the lines (1) and (2) are equally inclined to line (3) which means that the angle between lines (1) and (3)equals the angle between lines (2) and (3).

|m1m31+m1m3|=|m2m31+m2m3|

Substitute the values,

|3m1+3m|=|12m1+12m|

Take LCM

|3m1+3m|=|12mm+2|

It can be written as,

3m1+3m=±(12mm+2)

Here,

3m1+3m=12mm+2 or 3m1+3m=(12mm+2)

If 3m1+3m=12mm+2

By cross multiplication,

(3m)(m+2)=(12m)(1+3m)

m2+m+6=1+m6m2

5m2+5=0

Divide the equation by 5

m2+1=0

m=1, which is not real.

Therefore, this case is not possible.

If 3m1+3m=(12mm+2)

By cross multiplication,

(3m)(m+2)=(12m)(1+3m)

m2+m+6=(1+m6m2)

7m22m7=0

Here we get,

m=2±44(7)(7)2(7)

m=2±21+4914

Rewrite as,

m=1±527

Thus, the required value of m is 1±527.


19.If sum of the perpendicular distance of a variable point P(x,y) from the lines x+y5=0 and 3x2y+7=0 is always 10. Show that P must move on a line.

Ans.Given that,

x+y5=0(1)

3x2y+7=0(2)

Here the perpendicular distances of P(x,y) from lines (1) and (2)are written as,

d1=|x+y5|(1)2+(1)2 and d2=|3x2y+7|(3)2+(2)2

Nw, d1=|x+y5|2 and d2=|3x2y+7|13

We know that d1+d2=10

Substitute the values,

|x+y5|2+|3x2y+7|13=10

13|x+y5|+2|3x2y+7|1026=0

It can be written as,

13(x+y5)+2(3x2y+7)1026=0

Assume (x+y5) and (3x2y+7) are positive,

13x+13y513+32x22y+721026=0

Take out the common terms,

x(13+32)+y(1322)+(725131026)=0, which is the equation of a line.

Similarly, we can find the equation of line for any signs of (x+y5) and 

(3x2y+7).

Therefore, point P must move on a line.


20. Find equation of the line which is equidistant from parallel lines 9x+6y7=0 and 3x+2y+6=0.

Ans.The equation of the given lines are 9x+6y7=0(1)

3x+2y+6=0(2)

Consider P(h,k) be the arbitrary point that is equidistant from lines (1) and (2).

Here, the perpendicular distance of P(h,k) from line (1) is,

d1=|9h+6k7|(9)2+(6)2

=|9h+6k7|117

=|9h+6k7|313

Similarly, the perpendicular distance of P(h,k) from line (2) is,

d2=|3h+2k+6|(3)2+(2)2

=|3h+2k+6|13

We know that P(h,k) is equidistant from lines (1) and (2)

That is, d1=d2

Substitute the values,

|9h+6k7|313=|3h+2k+6|13

|9h+6k7|=3|3h+2k+6|

|9h+6k7|=±3(3h+2k+6)

Now, 9h+6k7=3(3h+2k+6) or 9h+6k7=3(3h+2k+6)

9h+6k7=3(3h+2k+6) is not possible as,

9h+6k7=3(3h+2k+6)

7=18 ,which is wrong

We have,

9h+6k7=3(3h+2k+6)

By multiplication

9h+6k7=9h6k18

18h+12k+11=0

Therefore, the required equation of the line is 18x+12y+11=0.


21. A ray of light passing through the point (1,2) reflects on the x-axis at point A and the reflected ray passes through the point (5,3). Find the coordinates of A.

Ans. 


Consider the coordinates of point


Consider the coordinates of point A as (a,0)

Construct a line (AL)which is perpendicular to the x-axis

Here, the angle of incidence is equal to angle of reflection.

That is,

BAL=CAL=

CAX =θ

It can be written as,

OAB=180(θ+2ϕ)=180[θ+2(90θ)]

=180θ180+2θ

=θ

Now, BAX=180θ

Slope of line AC=305a

tanθ=35a(1)

Slope of line AB=201a

tan(180θ)=21a

tanθ=21a

tanθ=2a1

From equations (1) and (2),

35a=2a1

By cross multiplication,

3a3=102a

a=135

Therefore, the coordinates of point A are (135,0).


22. Prove that the product of the lengths of the perpendiculars drawn from the points (a2b2,0) and (a2b2,0) to the line xacosθ+ybsinθ=1 is b2.

Ans.Given that,

xacosθ+ybsinθ=1

Rewrite as,

bxcosθ+aysinθab=0(1)

Here, the length of the perpendicular from point (a2b2,0) to line (1) is,   

p1=|bcosθ(a2b2)+asinθ(0)ab|b2cos2θ+a2sin2θ

=|bcosθa2b2ab|b2cos2θ+a2sin2θ

Similarly, the length of the perpendicular from point (a2b2,0) to line (2) is,

p2=|bcosθ(a2b2)+asinθ(0)ab|b2cos2θ+a2sin2θ

=|bcosθa2b2+ab|b2cos2θ+a2sin2θ

Multiply equations (2) and (3),

=|(bcosθa2b2ab)(bcosθa2b2+ab)|(b2cos2θ+a2sin2θ)

From the formula,

=|(bcosθa2b2)2(ab)2|(b2cos2θ+a2sin2θ)

Square the numerator,

=|b2cos2θ(a2b2)a2b2|(b2cos2θ+a2sin2θ)

Expand using formula,

=|a2b2cos2θb4cos2θa2b2|b2cos2θ+a2sin2θ

Take out the common terms,

=b2|a2cos2θb2cos2θa2|b2cos2θ+a2sin2θ

=b2|a2cos2θb2cos2θa2sin2θa2cos2θ|b2cos2θ+a2sin2θ

Here, sin2θ+cos2θ=1 (trigonometric identity)

=b2|(b2cos2θ+a2sin2θ)|b2cos2θ+a2sin2θ

=b2(b2cos2θ+a2sin2θ)(b2cos2θ+a2sin2θ)

Cancel common terms,

=b2

Hence, proved that the product of the lengths of the perpendiculars drawn from the points (a2b2,0) and (a2b2,0) to the line xacosθ+ybsinθ=1 is b2.


23. A person standing at the junction (crossing) of two straight paths represented by the equation 2x3y+4=0 and 3x+4y5=0 wants to reach the path whose equation is 6x7y+8=0in the least time .Find equation of the path that he should follow.

Ans.Given that,

2x3y+4=0(1)

3x+4y5=0(2)

6x7y+8=0(3) 

Here, the person is standing at the junction of the paths represented by lines 

(1) and (2).

Solve equations (1) and (2),

x=117 and y=2217

Thus, the person is standing at point (117,2217).

It is known that the person can reach path (3)in the least time if he walks along the perpendicular line to (3) from point (117,2217)

Here, the slope of the line (3)=67

Now, the slope of the line perpendicular to line (3)=1(67)=76

So the equation of line passing through point (117,2217)and having a slope of 76 is,

(y2217)=76(x+117)

6(17y22)=7(17x+1)

By multiplication,

102y132=119x7

119x+102y=125

Therefore, the path that the person should follow is 119x+102y=125.


NCERT Solutions for Class 11 Maths chapter 9 Sub-Topics

To help you have an initial idea of Ch 10 Maths Class 11, we provide here a brief overview of the concepts discussed in the chapter.


1. Part 1: Introduction

The first subsection introduces students to the concept of straight lines and their different parts and forms. Students will learn about identifying the coordinates of a point and measure the given lines with their help.


In Straight Lines Class 11 NCERT Solutions, you will find solutions to problems that require drawing shapes on a graph with provided coordinates, figuring out coordinates of a given shape’s vertices, and calculating the lengths of sides.


2. Part 2: Slope of a Line

This part is further divided into four sub-parts that delve into the intricacies of calculating the slope of a line. The first sub-topic explains the calculation of slope with given coordinates of two points on a line. Students are then introduced to the conditions for perpendicular and parallelism in terms of slope, calculating angles between two lines, and the conditions for collinearity of three points.


3. Part 3: Various Forms of the Equation of a Line

Here, you will learn about the different types of lines, vertical and horizontal. Additionally, there are 5 subsections dedicated to elaborate on each form of linear equation, namely, normal form, slope-intercept form, point-slope form, two-point form, and intercept-form.


4. Part 4: General Equation of a Line

This section in Straight Lines Class 11 introduces students to the general form of a linear equation and other equations of the same line.


You will find step-by-step answers on conversions of equations from one form to another in NCERT Solutions Class 11 Maths chapter 9.


5. Part 5: Distance of a Point from a Line

The formula to calculate the distance between a point and a straight line and the distance between two parallel lines have been explained in the last section. Straight Lines Class 11 NCERT Solutions PDF download is free on Vedantu. It comes with a comprehensive breakdown of numerical questions based on the application of these formulae that will help students understand the process in detail


Overview of Deleted Syllabus for CBSE Class 11 Maths Straight Lines

Chapter

Dropped Topics









Straight Lines

9.2.4 Collinearity of Three Points 

(Examples 4–5 and Question 8, 13–14 in Exercise 9.1)

9.3.6 Normal Forms

Exercise 9.2 - 8th Question

9.4 General Equation of a Line

Exercise 9.3 - 3rd Question

Miscellaneous Exercise - 2nd Question

Fourth Last Point in the Summary

9.6 Equation of Family of Lines Passing Through the Points of Intersection of Two Lines

9.7 Shifting of Origin



Class 11 Maths Chapter 9: Exercises Breakdown

Exercise

Number of Questions

Exercise 9.1

11 Questions & Solutions

Exercise 9.2

19 Questions & Solutions

Exercise 9.3

17 Questions & Solutions

Miscellaneous Exercise

23 Questions & Solutions



Conclusion

Chapter 9 Class 11 Maths is important for mastering the basics of coordinate geometry. It covers essential concepts such as the slope of a line, various forms of line equations, and the angles between lines. Focus on understanding these concepts deeply, as they form the foundation for more advanced topics in mathematics. From previous year question papers, about 4–6 questions were typically asked, highlighting the importance of this chapter in exams. With Vedantu’s comprehensive solutions, you can confidently solve these problems and excel in your studies.


Other Study Material for CBSE Class 11 Maths Chapter 9



Chapter-Specific NCERT Solutions for Class 11 Maths

Given below are the chapter-wise NCERT Solutions for Class 11 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.



Important Related Links for CBSE Class 11 Maths

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FAQs on NCERT Solutions for Class 11 Maths Chapter 9 Straight Lines

1. What are the topics discussed in Chapter 10 Maths Class 11?

NCERT Solutions for Class 11 Maths Chapter 10 introduces the students to coordinate geometry, which is a very crucial topic in the Class 11 Maths syllabus. Students will be required to understand the graphical representation of straight lines and their algebraic forms. They will also learn the form and applications of various representations of a first-degree equation besides the general equation of a line. 

Calculation of the slope and the distance of an external point from a given line is discussed with relevant examples to help students solve problems based on these concepts efficiently.

2. What is the equation of a line written in general form?

There are various forms of an equation that a graphical line can be represented in. The first and most common form of a linear equation is the general equation.

For a line represented in two variables x and y of the first degree, the general equation is given as Ax + By + C = 0, where coefficients A and B cannot be 0. A, B, and C are constants belonging to the set of real numbers, and variables x and y represent the coordinates on the respective axes.

3. How to find the slope of a line?

In Mathematics, the slope or gradient of a line, denoted by the letter m, describes the steepness and the direction of that line. Ideally the slope equation of a straight line is a linear function represented as y = f(x) = mx + c.

The average slope of a line is a constant for the same line and is calculated as a ratio between the differences of y-coordinates of two points, the rise, to that of x-coordinates of those points, the run.

You can use the slope equation in determining the slope by marking two points on a line and noting their x and y-coordinates. Next, determine the vertical and horizontal changes between those two points. For the final value, divide the difference between the y-coordinates by that of the x-coordinates. There you have slope m for a given line.

4. How many Exercises are there in Chapter 10 Maths Class 11?

There are a total of 3 Exercises in Class 11 Maths Chapter 10 'Straight Lines'. There are 14 questions in Exercise 10.1, 20 questions in Exercise 10.2 and 18 questions in Exercise 10.3. These exercises consist of questions that explain the fundamentals of coordinate geometry using graphical representations. To learn more, you can go through the NCERT solutions prepared by the team of Vedantu for easy learning. 

5. How can I strengthen my concepts of Class 11 Maths Chapter 10?

Many students in Class 11 struggle to grasp the concepts of Mathematics. What they require is excellent study material on which they can rely for high grades. Along with NCERT exercises, students should practise various important questions and familiarise themselves with previous year papers in NCERT solutions. You will be able to maintain a solid grip on the subject if you clear your concepts and practise them regularly.

6. Is Chapter 10 Maths Class 11 important?

Yes, Chapter 10 Maths Class 11 is an important chapter and students should not neglect this chapter at any cost. It comprises 13% of the marks weightage in the exams. Leaving this chapter may be a bad idea for you. So, to prepare for chapter 10 thoroughly, you can take the help of the study guides provided on the website of Vedantu. For easier access, you can download the Vedantu app.

7. Can I download NCERT Solutions for Chapter 10  Maths Class 11?

Absolutely yes. You can download the NCERT solutions of Class 11 Maths Chapter 10 from the website of Vedantu or you can also download the Vedantu app to get easy access through your phone. These solutions are in downloadable PDF format and are totally free of cost. Once downloaded, you can study from your device anytime and anywhere. 

8. How can I benefit from using Vedantu?

Choosing Vedantu brings you a load of advantages. Following are some benefits that you can avail yourself by using Vedantu:

  • It provides NCERT solutions to all the questions prepared by our professional tutors with years of experience. 

  • Prepared notes and important questions are provided after extensive research and study.

  • Improves your grades as well as prepares you for competitive exams.

  • Provided study guides that strictly adheres to the CBSE curriculum.

9. What is the significance of understanding the slope of a line in class 11 maths chapter 9 solutions?

The slope of a line is a fundamental concept that indicates how steep a line is and its direction. Understanding the slope is crucial for solving problems related to parallel and perpendicular lines. It also helps in determining the angle of inclination of the line with respect to the x-axis. The slope is widely used in various applications, including physics, engineering, and computer graphics in class 11 maths chapter 9 solutions.

10. How does chapter 9 help in real-world applications in ncert straight lines class 11 solutions?

The concepts learned in ncert straight lines class 11 solutions, such as finding the equation of a line and calculating distances, are vital in fields like engineering, physics, and computer science. They help in designing and analysing structures, understanding motion and trajectories, and solving practical problems involving straight lines.

11. How many questions from chapter 9 were asked in previous year exams in chapter 9 class 11 maths?

On average, about 4–6 questions from chapter 9 class 11 maths have been asked in previous year exams. These questions often cover key topics like the slope, equations of lines, and angles between lines. This indicates the chapter's significance in the Class 11 Maths syllabus and highlights the need for thorough understanding and practice to perform well in exams.