Straight Lines

Straight Lines Class 11

Coordinate geometry refers to the field of mathematics that deals with a geometrical study using algebra. A French mathematician Rene Descartes realised that representation of a curve or a straight line is possible using an algebraic equation. In this field of mathematics, an ordered pair of numbers (real numbers) represents a point in a plane and is known as coordinates of that point. Also, a curve or a straight line is illustrated using an algebraic equation having real coefficients.

Notably, this is one of the most important chapters since numerous other functions and calculations also depend on it.

What is Meant by a Straight Line?

A straight line is a line traced by a point which is travelling in a fixed direction having zero curvature. In simpler words, straight lines can also be said as the shortest paths between two points.

Straight Line: Slope

  • Inclination

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When a straight line L makes an angle with +ve x-axis which is evaluated in anticlockwise direction till the portion of the line L above the x-axis is known as angle of inclination or just inclination of the line. Usually, it is denoted by theta (θ). According to the figure shown above, it is evident that theta is higher than equal to 0 degrees and less than 180 degrees.

Among vital pointers to note,

  1. The angle of inclination of any line which is parallel to x-axis or x-axis itself is 0 degree.

  2. The angle of inclination of any line which is parallel to y-axis or y-axis itself is 90 degrees.

  • Vertical, Horizontal and Oblique Lines

  1. A line which is parallel to x-axis or x-axis itself is termed as a straight horizontal line.

  2. A line which is parallel to y-axis or y-axis itself is termed as a vertical straight line.

  3. A line which is neither vertical nor horizontal is termed as an oblique line.

  • Slope or Gradient of a Straight Line

If theta (not equal to 90 degrees) is the angle of inclination of all straight lines, then their slope or gradient is tan θ. It is represented by “m”.

Therefore, it can be said that when θ (not equal to 90 degrees) is an angle of inclination, m is equal to tan θ.

Observations:

  1. As tan θ is not determined when θ is 90 degrees, a vertical line’s slope is also not determined.

  2. Y-axis slope is not determined.

  3. The angle of inclination of all parallel lines to the x-axis is 0 degree, so the slope is tan 0 degree which is equal to 0. Hence, every horizontal line’s slope is 0.

  4. X-axis slope is 0.

How to Find a Straight Line’s Slope Which Joins Two Points?

The slope in a straight line (which is non-vertical) moving through two points can be determined by the following expression.

Consider P(x1, y1) and Q(x2, y2) are two points on line l, which is non-vertical. As the line is non-vertical, then x2 is not equal to x1.

Let the angle of inclination of line l be θ, and it can be obtuse or acute. Both cases are required to be considered. From Q and P, perpendiculars QN and PM on x-axis and PL perpendicular to NQ are drawn, which is shown the following figures:

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Case 1. When θ is acute:

PL= MN= ON – OM = x2 – x1 and LQ= NQ – NL = NQ – MP = y2 – y1

So, angle QPL = θ

Therefore, tan θ = LQ/PL = y2 –y1 /x2 –x1

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Case 2. When θ is obtuse:

LP= NM = OM-ON = x1 – x2 and LQ= NQ – NL = NQ – MP = y2 – y1

So, angle QPL = π – θ

Therefore, tan (π – θ) = LQ/LP = y2 – y1 /x1 – x2

-tan θ = y2 – y1/x1 – x2

tan θ = y2 – y1 / x2 – x1

So, for both the cases, slope m is same.

The above concise information related to straight lines will be able to clear your concept about the slope of a straight line. As this chapter comprises of various line equations and types of straight lines, you must refer to our online study materials and download the Vedantu app for online classes.

FAQ (Frequently Asked Questions)

1. What is the Slope of a Line Moving through Points (i) (3, -2) and (-1, 4) (ii) (3, -2) and (7, -2)?

Ans. (i) By substituting the values in the equation, the slope of line moving through (3, -2) and (-1, 4) is

= (4 -(- 2)) / (-1 – 3)

= 6 / -4

= -3 / 2

(ii) Now for the points (3,-2) and (7, -2) the slope is:

= (-2 -(-2)) / (7-3)

= 0 / 4

= 0

2. What is the Value of k if Three Straight Lines 2x + y – 3= 0, 5x + ky -3 =0 and 3x – y – 2=0 are Concurrent?

Ans. Since the three lines are said to be concurrent, the intersecting point of two lines falls on the third line.

Let the given equations be marked as:

2x + y -3 = 0.........(1)

5x + ky – 3 = 0 ……(2)

3x – y – 2 = 0 ……..(3)

Solving equations 1 and 3 by cross multiplication procedure:

x/-2-3 = y/ -9+4 = 1/-2-3 or x=1 and y=1

Hence, the intersecting point of the two straight lines is (1,1). Now, by putting the values in equation 2:

5 . 1 + k . 1 – 3 = 0 or k = -2

The value of k is -2

3. What will be the Straight-line Equation of Three Points (5,1), (1,-1) and (11,4)? Also, Prove that these Points on the Straight Line are Collinear.

Ans. Let the mentioned points be P(5,1), Q(1, -1) and R(11,4). So, the line equation is:

y-1 = (-1 -1/1-5) (x-5)

y-1 = (-2/-4) (x-5)

y-1 =1 /2 (x-5)

2(y-1) = (x-5)

2y -2 = x-5

x- 2y – 3 =0

Now it is seen that R(11,4) satisfies x – 2y – 3 = 0. Therefore it can be said that all the points lies on the same line and they are collinear.

4. What will be the Fourth Vertex if Three Back to Back Vertices of a Parallelogram are (-2,-1), (1,0) and (4,3)?

Ans. Consider the vertices to be A(-2,-1), B(1,0), C(4,3) and D(x,y). Then the mid-point of AC will be (-2+4/2, -1+3/2) which is (1,1).

Moreover, the mid-point of BD will be ((1+x)/2, (0+y)/2).

As the parallelogram diagonals bisect each other, the midpoints are the same.

(1+x)/2 =1 and (0+y)/2 =1

Which means x = 1 and y = 2

Therefore the fourth vertex is (1,2).