Straight Lines Class 11 Notes - Chapter 10

The slope of a line (m) is determined by the value of tan θ, where, θ is the angle made by the line with the positive direction of x-axis in an anti-clockwise direction. The slope of the line passing through two points P (a1, b1) and Q (a2, b2) is given by:

$$\begin{array}{l}m\;=\;tan\;\theta \;=\;\frac{b_{2}-b_{1}}{a_{2}-a_{1}}\end{array}$$

If m1 and m2 be the slopes of two lines. The angle θ between them is given by:

$$\begin{array}{l}tan\;\theta =\;\pm \frac{(m_{1}-m_{2})}{1\;+\;m_{1}m_{2}}\end{array}$$

In case of acute angle,

$$\begin{array}{l}tan\;\theta =\;\pm \frac{(m_{1}-m_{2})}{1\;+\;m_{1}m_{2}}\end{array}$$
• For parallel lines, m1 = m2
• For perpendicular lines, m1.m2 = -1

Three points A(h, k), B(m1, n1) and C(m2, n2) are said to be collinear, if, the slope of AB = slope of BC.

i.e.

$$\begin{array}{l}\frac{n_{1}-k}{m_{1}-h}=\frac{n_{2}-n_{1}}{m_{2}-m_{1}}\end{array}$$
Or,
$$\begin{array}{l}(k-n_{1})\;(m_{2}-m_{1})\;=\;(h-m_{1})\;(n_{2}-n_{1})\end{array}$$

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Equation of Line – Different Forms

1. The equation of a line parallel to the x-axis and at a distance (p) from x-axis is given by, y = ± p.
2. The equation of a line parallel to the y-axis and at a distance (q) from y-axis is given by, x = ± q.
3. The equation of a line [Point-slope form] having slope (m) and passing through point (a0, b0) is given by, y – b0 = m(x – a0).
4. The equation of a line [Two-point-form] passing through two points (a1, b1) and (a2, b2) is given by,
$$\begin{array}{l}y-b_{1}=\left ( \frac{b_{2}-b_{1}}{a_{2}-a_{1}} \right )(x-a_{1})\end{array}$$

5. The equation of a line [Slope intercept form] making an intercept (p) on the y-axis (slope m) is given by y = mx + p. [value of p will be +ve or -ve based on the intercept made on
the +ve or -ve side of the y-axis].

1. The equation of the line [Intercept form] making intercepts p and q on x and y-axis respectively is given by
$$\begin{array}{l}\frac{x}{a}+\frac{y}{b} =1\end{array}$$

In normal form, the equation of the line is given by x cos ω + y sin ω = p. Where, p = Length of perpendicular (p) from the origin and ω = Angle which normally makes with the +ve x-axis direction.

The points (m1, n1) and (m2, n2) are on the same or opposite side of a line px + qy + r = 0, if pm1 + qn1 + r and pm2 + qn2 + r are of the same sign or of opposite signs respectively. The lines xm1 + yn1 + o1 = 0 and xm2 + yn2 + o = 0 are perpendicular, if, m2m1 + n2n1 = 0.

Straight Lines Class 11 Practice Questions

1. Determine the equation of lines passing through (2, 3) and making angle 60° with
x-axis.
2. Determine the equation of a line parallel to the line 2x + 5y = 2 and passing through the point of intersection of lines 12x + 2y = 35 and 4x + 53y + 68 = 0.
3. Determine the length of the triangle if the equation of the base of an equilateral triangle is 2x + 3y = 42 and the vertex is (3, -4).
1. Determine the value of p and q if the intercepts cutoff on the coordinate axes by the
line px + qy + 3 = 0 are equal but have opposite signs to those cut off by the
line x – 2y + 4 = 0 on the axes.
1. Determine the equation of the line if the intercept of a line connecting the coordinate axes is separated by a point (4, -3) in the ratio 2:4.

Frequently asked Questions on CBSE Class 11 Maths Notes Chapter 10: Straight Lines

What are ‘Straight lines’?

In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth.

What is ‘Distance formula’?

The distance formula to calculate the distance between two points

What is ‘Concurrency’?

Concurrency is the concept of executing two or more tasks at the same time.