NCERT Exemplar for Class 11 Maths - Straight Lines - Free PDF Download
Free PDF download of NCERT Exemplar for Class 11 Maths Chapter 10 - Straight Lines solved by expert Maths teachers on Vedantu.com as per NCERT (CBSE) Book guidelines. All Chapter 10 - Straight Lines exercise questions with solutions to help you to revise the complete syllabus and score more marks in your examinations.
NCERT Exemplar for Class 11 Maths Chapter 10 - Straight Lines explains the concept of equations of straight lines in different formats and how to calculate these equations through a graph. This chapter also explains the types of slopes and the intercepts (x-axis and y-axis) and different properties of straight lines followed by exercises related to the same.






Access NCERT Exemplar Solutions for CBSE Class 11 Mathematics Chapter 10: Straight Lines
Example 1: Find the equation of a line which passes through the point
Ans: Given: Line passes through
The slope of line
According to the point-slope form, the equation of a line with slope
Therefore, the equation of the line is
Example 2: Find the equation of a line where length of the perpendicular segment from the origin to the line is
Ans: Given: Length of perpendicular segment from origin to line is
normal form of the line is
Using
Therefore, the equation of the line is
Example 3: Prove that every straight line has an equation of the form
Ans: Given: Every straight line has an equation of the form
A straight line either cuts the
Hence verified.
Example 4: Find the equation of the straight line passing through
Ans: Given: The straight line passes through
Let the slope of the required line be
The slope of given line
As the two lines are perpendicular, the product of their slopes will be equal to
So, the equation of required line is
Therefore, the equation of the line is
Example 5: Find the distance between the lines
Ans:Given:
The distance between the parallel lines
The given lines
So, the distance between them is
Therefore, the distance between the two lines is
Example 6: Show that the locus of the mid-point of the distance between the axes of the variable line
Ans:
Given:
The intercept form of the line
Let the midpoint of the line segment joining
So,
Square both sides of equation and add them.
Therefore, the locus is
Example 7: If the line joining two points
The slope of the line joining
After rotating the line

So, the equation of new line will be
Therefore, the equation of line is
Example 8: If the slope of a line passing through the point
Ans: Given:Slope of a line passing through
The equation of line passing through
So,
Substituting
Substitute these values in
Therefore, the points are either
Example 9: Find the equation to the straight line passing through the point of intersection of the lines
Ans: Given: Line passes through point of intersection of
The equation of line passing through intersection of
Slope of this line is:
Slope of the line
The product of slopes of perpendicular lines will be
So,
So, the equation of required line will be
i.e.
Therefore, the equation of line is
Example 10: Aray of light coming from the point
Ans: Given: Theray of light comes from
Let the incident ray strike

Slope of reflected ray is:
Slope of incident ray is
From
Therefore, the coordinates of
Example 11: If one diagonal of a square is along the line
Ans: Given:
Let the square be

Let the slope of
So, equation of
The equation of
Therefore, the equations are
Example 12: The inclination of the line
(A)
(B)
(C)
(D)
Ans: Given:
Therefore,
Correct Option: A
Example 13: The two lines
(A)
(B)
(C)
(D)
Ans: Given:
Slope of
And slope of
The lines will be perpendicular if
Therefore,
Correct Option: A
Example 14: The equation of the line passing through
(A)
(B)
(C)
(D)
Ans:
Given:
Let the slope of line be
As the line passes through
The lines will be perpendicular if
So, the equation of line will be
Therefore,
Correct Options : B
Example 15: The distance of the point
(A)
(B)
(C)
(D) None of these
Ans: Given:
The distance of
Therefore, the distance is
Correct Options: C
Example 16: The coordinates of the foot of the perpendicular from the point
(A)
(B)
(C)
(D)
Ans: Given:
Let
Slope of perpendicular line
Slope of
The lines will be perpendicular if
As
Solving
Therefore,
Correct Option: B
Example 17: The intercept cut off by a line from
(A)
(B)
(C)
(D)
Ans:
Given: Line passes through the point

Then intercept on
So, equation of line is
As the line passes through
So, the equation of line will be
Therefore,
Correct Option : A
Example 18: A line passes through the point
(A)
(B)
(C)
(D)
Ans: Given: Line passes through the point
Let the intercept made by the line be
So, equation of line is
As it is bisected at
So, the equation of line will be
Therefore,
Correct Option: D
Example 19: The reflection of the point
(A)
(B)
(C)
(D)
Ans: Given:

Let
The midpoint of line joining
This point is on
So,
Slope of line joining
Solving
Therefore,
Correct Option: A
Example 20: A point moves such that its distance from the point
(A)
(B)
(C)
(D) None of these
Ans: Given:
Let
Given that the point moves in such a way that its distance from

So,
Therefore,
Correct Option: A
Short Answer Type Questions
EXERCISE 10.3:
1. Find the equation of a line which passes through the point
Ans:
Given: Line passes through the point
Let the intercept made by the line be
So, equation of line is
Given that
It passes through
So, equation of line will be
Therefore, the equation is
2. Find the equation of a line which passes through the point
Ans:
Given:
Slope of line joining
Slope of the line perpendicular to this line is
The equation of line that passes through
Therefore, the equation is
3.Find the angle between the lines
Ans:
Given:
Slope of line
Slope of the line
Using
Therefore,
4.Find the equation of lines which passes through the point
Ans:
Given: Line passes through the point
Let the intercept made by the line be
So, equation of line is
Given that
So,
This equation passes through
So, the equation of line when
The equation of line when
Therefore, the equations are
5. Find the points on the line
Ans:
Given:
Let the point
The distance of
Take the
From
Put this value in
So, the point is
Take the
From
Put this value in
So, the point is
Therefore, the points are
6. Show that the tangent of an angle between the lines
Ans:
Given: The tangent of an angle between
Given that
Slope of the line
Slope of the line
Using
Hence verified.
7. Find the equation of lines passing through
Ans:
Given: lines passing through
The line makes
So, the angle made with
Slope of line
The equation of line that passes through
Therefore, the equation is
8. Find the equation of the line passing through the point of intersection of
Ans:
Given: Line passes through the point of intersection of
The equation of line that passes through the point of intersection of
Slope of this line is
Slope of
As the two lines are parallel,
The equation
Therefore, the equation is
9. For what values of
Ans:
Given:
We can write
The intercepts are
We can write
The intercepts are
From the question,
Therefore,
10. If the intercept of a line between the coordinate axes is divided by the point
Ans:
Given:
Assume
Then, the coordinates are
Using
So,
And
So,
The equation of line
Therefore,
11. Find the equation of a straight line on which length of perpendicular from the origin is four units and the line makes an angle of
Ans:
Given: Line makes
The line makes
So, slope of the line is
Equation of the required line is:
This line is 4 units away from
Therefore, the equation of line is
12. Find the equation of one of the sides of an isosceles right angled triangle whose hypotenuse is given by
Ans:
Given:
The hypotenuse is along the line
Slope of the hypotenuse is
As the triangle is isosceles right angled triangle, two angles will be equal to
Let the slope of the line making
Slope of this line is
The slope of line perpendicular to it is
The equation of this line is:
Therefore, the equation of required line is
13. If the equation of the base of an equilateral triangle is
Ans:
Given:
The altitude from the vertex
The length of altitude is the distance of
The triangle is an equilateral triangle. So,
Therefore, the length of side of the triangle is
14. A variable line passes through a fixed point
Ans:
Given:
Assume that the variable line passing through the fixed point
Perpendicular distance from the point
Perpendicular distance from the point
Perpendicular distance from the point
Given that,
Therefore, the variable line passes through the fixed point
15. In what direction should a line be drawn through the point
Ans:
Given:
Let the slope of line be
It passes through
So, the equation of line is
The equation of the given line is
Assume that these lines meet at point
Solving
Given that
Therefore,
16. A straight line moves so that the sum of the reciprocals of its intercepts made on axes is constant. Show that the line passes through a fixed point.
Ans:
Given:
The equation of line in intercept form is
As
Let
So,
Thus,
Therefore, the line passes through the fixed point
17. Find the equation of the line which passes through the fixed point and the portion of the line intercepted between the axes id divided internally in the ratio 5:3 by this point.
Ans: Let AB be a line passing through a point (- 4, 3) and meets x-axis at A(a, 0) and y-axis at B(0, b).

Intercept form of the line is
Hence, The required equation is
18. Find the equation of the lines through the point of intersection of the lines and and whose distance from the point is
Ans: Given equation are
we get
From eq. (i) we have
So,
Let m be the slope of the required line
Since, the perpendicular distance from
to the line is
Equation of the line taking
Equation of the line taking
Hence, the required equation are
19.If the sum of the distance of a moving point in a plane from the axes is 1, then find the locus of the point.
Ans: Let coordinates of a moving point P be
Given that the sum of the distance from the axes to the point is always 1

Hence, these equations gives us the locus of the point P which is a square.
20. P1, P2 are points on either of the two lines at a distance of 5 units from their point of intersection. Find the coordinates of the foot of perpendiculars drawn from P1, P2 on the bisector of the angle between the given lines.

Ans: Given lines are
And
Slope of eq. (i) is
Slope of eq. (ii) is
120° Solving eq. (i) and eq. (ii) we get
Putting the value of y is eq. (i) we get
Point of intersection of line (i) and (ii) is
Hence, the coordinates of the foot of perpendicular
21.If p is the length of perpendicular from the origin on the line and are in A.P. then show that
Ans: Given equation is
Since,
Squaring both sides, we have
Putting the value of
Hence proved.
22. A line cutting off intercept
(A)
(B)
(C)
(D) None of these
Ans:
Given: A line cuts off an intercept
Let the equation of required line be
As
The equation of line is:
Hence, the equation is
Correct Option : A
23. Slope of a line which cut off intercepts of equal lengths on the axes is
(A)
(B)
(C)
(D)
Ans:
Given: A line cuts off intercepts of equal lengths on the axes.
The equation of required line is
Slope
Hence, the slope is
Correct Option: A
24. The equation of the straight line passing through the point
(A)
(B)
(C)
(D)
Ans:
Given:
Slope of
So, the slope of line perpendicular to it is
The line passes through
So, equation of required line is
Hence, equation is
Correct Option: B
25. The equation of the line passing through the point
(A)
(B)
(C)
(D)
Ans:
Given:
Slope of
So, the slope of line perpendicular to it is
The line passes through
So, equation of required line is
Hence, equation is
Correct Option: B
26. The tangent of angle between the lines whose intercepts on the axes are
(A)
(B)
(C)
(D) None of these
Ans:
Given: The intercepts on the axes are
The intercepts of line are
So, the line passes through the points
Thus, the slope of line is
The intercepts of line are
So, the line passes through the points
Thus, the slope of line is
Let
Hence,
Correct Option: C
27. If the line
(A)
(B)
(C)
(D)
Ans:
Given:
The points
And
Multiply
Put
So,
Hence,
Correct Option: D
28. The distance of the point of intersection of the lines
(A)
(B)
(C)
(D) None of these
Ans:
Given:
Solve the lines
The point of intersection is
The distance of
The distance of
The distance of
Hence, the distance is
Correct Option: A
29. The equations of the lines which pass through the point
(A)
(B)
(C)
(D) None of these
Ans:
Given:
The slope of
Let the slope of required line which makes
Then,
Line passes through
So, the equation of line is:
Hence, the equations are
Correct Option: A
30. The equations of the lines passing through the point
(A)
(B)
(C)
(D) None of these.
Ans:
Given: Line passes through
Let the slope of the required line be
The equation of line passing through
The distance of this line from
Hence, equations of lines are:
Correct Option: A
31. The distance between the lines
(A)
(B)
(C)
(D) 0
Ans:
Given:
Let
The other line is:
Distance of
Hence, distance is
Correct Option: B
32. The coordinates of the foot of perpendicular from the point
(A)
(B)
(C)
(D)
Ans:
Given:
Let the foot of perpendicular from
The slope of the given line is
Slope of the line joining
Solve
Hence, the coordinates of foot of perpendicular is
Correct Option: B
33. If the coordinates of the middle point of the portion of a line intercepted between the coordinate axes is
(A)
(B)
(C)
(D)
Ans:
Given:
As the middle point is
So, the equation of the line using intercept form is:
Hence, the equation is
Correct Option: A
34. Equation of the line passing through
(A)
(B)
(C)
(D)
Ans:
Given:
The required line is parallel to
Thus, slope of the line is
The line also passes through
So, the equation of line is:
Hence, the equation is
Correct Option: C
35.Equations of diagonals of the square formed by the lines
(A)
(B)
(C)
(D)
Ans:
Given:
Using the equations
The equation of
The equation of
Hence, the diagonals are
Correct Option: A
36. For specifying a straight line, many geometrical parameters should be known?
(A)
(B)
(C)
(D)
Ans:
Given: A straight line is specified.
The general equation of a linear equation in two variables is
At least one of
Assume
Then, the equation of the line is:
Hence, two parameters should be known.
Correct Option: B
37. The point
(i) Reflection about the line
(ii) Translation through a distance
(A)
(B)
(C)
(D)
Ans:
Given:
The reflection of
Then, translation of
Hence, the final coordinates of point is
Correct Option: B
38. A point equidistant from the lines
(A)
(B)
(C)
(D)
Ans: Given:
The distance of
The distance of
The distance of
The distance of all the three lines from
Hence,
Correct Option: C
39. A line passes through
(A)
(B)
(C)
(D)
Ans:
Given: Line passes through
The slope of
So, slope of a line perpendicular to it
The equation of the required line is:
Put
Hence, the
Correct Option: D
40. The ratio in which the line
(A)
(B)
(C)
(D)
Ans: Given:
Distance between
Distance between
Hence, the ratio is
Correct Option: B
41. One vertex of the equilateral triangle with centroid at the origin and one side as
(A)
(B)
(C)
(D)
Ans: Given:
Let
The centroid is
Slope of line
As
The slope of
The distance of
Distance of
Hence, the vertex is
Correct Option:C
42. If
Ans: Given:
So,
Compare
Hence, the point
43. The line which cuts off equal intercept from the axes and pass through the point
Ans: Given:
The line which cuts equal intercepts from the axes is
As this line passes through
Thus, the required equation of line is:
Hence, the equation is
44. Equations of the lines through the point
Ans: Given: Line passes through
Slope of
Let the slope of the required line be
The line also passes through
The equation of the line is:
Hence, equations of lines are:
45. The points
Ans: Given:
Given line is
For the point
For the point
Hence,
46. A point moves so that square of its distance from the point
Ans: Given:
Let the moving point be
Given point is
The distance of
Given that,
Hence, locus is:
47. Locus of the mid-points of the portion of the line
Ans: Given:
Let
By squaring and adding,
Hence, the locus is
State whether the Statements in Exercises
48. If the vertices of a triangle have integral coordinates, then the triangle cannot be equilateral.
Ans:The given statement is true.
Let us assume an equilateral triangle with vertices
Here
Therefore, in expression
49. The points
Ans: The given statement is false.
Using the distance formula, the length
Similarly, the length
And, the length
Clearly the sum of any two line segments is not equal to the third one in any case, so the points are not collinear.
50. Equation of the line passing through the point
Ans:The given statement is false.
The slope of the given line
Therefore, the slope of the required line which is perpendicular to the given line is,
So the equation of the line passing through the point
Using the algebraic identity
Using the identity
51. The straight line
Ans: The given statement is true.
The intersection of the lines
Now, if the line
Hence, the point satisfies the line.
52. The vertex of an equilateral triangle is
Ans: The given statement is true.
Let us assume the slope of one of the remaining two sides as
The angle between any two sides of an equilateral triangle is
Removing the modulus sign,
Considering the positive sign,
Considering the negative sign,
Hence, there are two values of
53. The equation of the line joining the point
Ans: The given statement is true.
Solving the equations
The equation of the line passing through the points
Now, the distance of the point
Similarly, the distance of the point
Hence, the points are equidistant from the required line.
54. The lines
Ans: The given statement is true.
Let the coordinates of the foot of perpendicular drawn from the origin
The given line is
The foot of perpendicular is given as,
Using the given relation
Equating the relations
and
On squaring and adding,
55. The lines ax + 2y + 1 = 0 , bx + 3y + 1 = 0 and cx + 4y + 1 = 0 are concurrent if a, b and c are in G.P.
Ans: The given statement is false.
The given lines can be written are
$\Rightarrow\left|\begin{array}{lll}
a & 2 & 1 \\
b & 3 & 1 \\
c & 4 & 1
\end{array}\right|=0$
Performing the row operations
$\Rightarrow\left|\begin{array}{ccc}
a & 2 & 1 \\
b-a & 1 & 0 \\
c-a & 2 & 0
\end{array}\right|=0$
Expanding the determinant along
Therefore, for the lines to be concurrent
56. Line joining the points
Ans: The given statement is false.
The line joining the points
The slope of this line is
Similarly, the line joining the points
The slope of this line is also
For the two lines to be perpendicular the product of their slopes must be
Match the questions given under Column
57.
Column | Column |
(a) The coordinates of the points are at a distance of | (i) |
(b) The coordinates of the points on the line from the line | (ii) |
(c) The coordinates of the points on the line joining that | (iii) |
Ans:
(a) Let one of the required point is
The distance of the point
Considering the positive sign and solving equation
Considering the negative sign and solving equation
(b) Let one of the required point is
The distance of the point
Considering the positive sign and solving equation
Considering the negative sign and solving equation
(c) Here the points
For point
Using this formula for the points
Now,
Using the midpoint formula for
Solving for the coordinates of
Hence, the correct match is (a) – (iii), (b) – (i), (c) – (ii).
58. The value of the
Column | Column |
(a) Parallel to | (i) |
(b) Perpendicular to | (ii) |
(c) Passes through | (iii) |
(d) Parallel to | (iv) |
Ans: Simplifying the line
gives
(a) For the line
(b) The slope of the line
(c) If the point
(d) For the line
Hence, the correct match is (a) – (iv), (b) – (iii), (c) – (i), (d) – (ii).
59. The equation of the line through the intersection of the lines
Column | Column |
(a) Through the point | (i) |
(b) Perpendicular to the line | (ii) |
(c) Parallel to the line | (iii) |
(d) Equally inclined to the axes is | (iv) |
Ans: The point of intersection of the lines
(a) The equation of the line passing through the points
(b) The slope of the line
(c) The slope of the line
(d) The lines which are equally inclined to the axes have their slope values equal to
Considering the positive slope,
Considering the negative slope,
Hence, the correct match is (a) – (iii), (b) – (i), (c) – (iv), (d) – (ii) or (iii).
About the Chapter
A straight line is defined as a combination of an infinite number of points joined at both ends of a point. In other words, it is an endless one-dimensional figure that does not have any curve but the ends can extend up to infinity. A straight line can be horizontal, vertical and slanted and have equations associated with the same. Any straight line, on a plane, will always be measured at 180° for an angle drawn between two points on the same straight line. A straight line can also be called the shortest distance between two points and is represented on the graph as A(x1, y1) and B(x2, y2). There are three kinds of straight lines:
A Horizontal Line is parallel to the x-axis and perpendicular to the y axis. The angle formed is 0° or 180° with the x-axis and 90° or 270° with the y axis on the graph.
A Vertical Line is parallel to the y axis and perpendicular to the x-axis. The angle formed is 90° or 270° with the x-axis and 0° or 180° with the y axis on the graph.
A Slanted Line can form any angle other than 0°, 90°, 180°, 270° and 360° with the horizontal and vertical lines on the graph.
Other details like equations of the slopes formed by the straight lines, equations of lines parallel to the x-axis or y-axis, various forms of the equation of a line, etc., have also been dealt with.
NCERT Exemplar for Class 11 Maths Chapter 10 - Straight Lines is divided into different sections from 10.1 to 10.1.8. Section 10.2 deals with examples on the different topics of straight lines and Section 10.3 comprises exercises/question bank.
FAQs on NCERT Exemplar for Class 11 Maths Chapter 10 - Straight Lines (Book Solutions)
1. What is the content of NCERT Exemplar for Class 11 Maths Chapter 10 - Straight Lines?
Chapter 10 Straight Lines in NCERT class 11 Maths has been explained conceptually so that the lesson is etched in the student’s memory without fail. Each topic has been concisely analysed and presented with examples and different methods to solve different questions. This chapter is split into several segments:
10.1- Overview of Straight Lines
10.1.1- Slope of a Line
10.1.2- Angle between two lines
10.1.3- Collinearity of three points
10.1.4- Various forms of the equation of a line
10.1.5- General equation of a line
10.1.6- Distance of a point from a line
10.1.7- Locus and Equation of a Locus
10.1.8- Intersection of two given lines
2. What are the various forms of the equation of a line?
The general equation of a straight line is ax+by+c=0 where a, b and c are constants, x and y are variables and the slope is -a/b. The various slopes associated with this equation are as follows:
Slope and Y-Intercept Form where slope m = tanθ. θ is the angle formed by the line with the positive x-axis, and y-intercept as b is given by: y = mx + b, where m is the slope.
Slope Point Form in which a straight line passes through a point where y-y1= m(x-x1)
Two Points Form in which a straight line passed through two points where (y-y1)= [(y2-y1)/x2-x1](x-x1)
Intercept Form in which one point A is on the y axis and point B is on the x-axis where (x/a)+(y/b)=1
3. What are the different kinds of slopes of a straight line?
There are four different kinds of slopes formed by a straight line. They are as follows:
Zero slopes in which the angle formed by the straight line is 0° with the x-axis, ie., it is parallel to the x-axis.
Positive Slope in which the angle formed with its x axis is between 0° and 90° hence the slope of the line is positive.
Negative Slope in which the angle formed with its x-axis is between 90° and 180° hence the slope of the line is negative
Infinite Slope in which the line forms a 90° angle with the x-axis or lies parallel to the y axis. Then the slope is not defined or is infinite.
4. Where can we download NCERT Exemplar for Class 11 Maths Chapter 10 - Straight Lines?
Since NCERT books are available online for free, students can easily access the book at ncert.nic.in and download the book in PDF format. Vedantu has also created study materials on NCERT Exemplar for Class 11 Maths Chapter 10 - Straight Lines (Book Solutions) with solved questions and examples for students to understand the concepts even better. Instead of searching on the internet for hours for resources and other study materials, students can click on Vedantu's official website and download the same for free.
5. Can solving the Previous Years’ Question Paper clear all doubts on Straight Lines?
Previous Year’s Question Papers are very important to solve since they help a student boost his/her confidence and also solving different levels of questions helps them enhance their performance. They are an important tool for revision that students can practice after solving the exercises in the NCERT textbook. It also helps the student keep a track of their strengths and weaknesses for improvement and provides a set pattern of the exam and the weightage of marks on each question.

















