Straight Lines Class 11 Notes Mathematics Chapter 10 - CBSE

Chapter : 10

What Are Straight Lines ?

Slope Of A Line

If θ is the inclination of a line l then tan θ is called the slope or gradient of the line l. The slope of the line is denoted by m.

Thus, m = tan θ, θ = 90

It may be observed that line slope of X-axis is zero and slope of Y-axis is not defined.

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Slope of a Line when coordinates of any two points on the Line are given:

In figure ∠MPQ = θ. Therefore, slope of line

l = m tan θ ...(i)

But ∆MPQ, we have

$$\text{tan}\space\theta =\frac{\text{MQ}}{\text{MP}}\\=\frac{\text{y}_2- \text{y}_1}{\text{x}_2 - \text{x}_1}\\\text{...(ii)}$$

From equation (i) and equation (ii), we have

$$m =\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$

Angle Between Two Line

The angle θ between the lines having slopes m1 and m2 is given by

$$\text{tan}\space\theta =\pm\frac{m_{1}-m_{2}}{1 + m_{1}m_{2}}$$

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The acute angle between the lines is given by

$$\text{tan}\space\theta =\begin{vmatrix}\frac{\text{m}_{1}- \text{m}_{2}}{1 + \text{m}_{1}\text{m}_{2}}\end{vmatrix}$$

The acute angle between the lines is given by

Condition of Parallelism of Lines

If two lines of slope m1 and m2 are parallel, then the angle θ between them is of 0°

∴ tan θ = tan 0° = 0

$$\frac{m_{1}-m_{2}}{\text{1 + m}_1\text{m}_2} = 0$$

m1 = m2

Thus, when two lines are parallel, their slopes
are equal.

Condition of Perpendicularity of Two Lines

If two lines of slopes m1 and m2 are perpendicular, then the angle θ between them is of 90°

∴ tan θ = ∞ or cot θ = 0

$$\frac{1 + m_{1}m_{2}}{m_1 - m_2 =-1} = 0$$

Thus, when two lines are perpendicular, the product of their slopes it -1. If m is the slope of a line, then the slope of a line perpendicular to it is $$\frac{\normalsize-1}{\text{m}}$$

Equation Of A Line Parallel To X-axis

Equation of a line parallel to X-axis at a distance b from it is y = b.

Equation Of A Line Parallel To Y-axis

Equation of a line parallel to Y-axis at a distance a from it is x = a.

Different Forms Of The Equation Of A Straight Line

Slope Intercept form of a Line

Equation of line with slope m and making an intercept c on Y-axis is

y = mx + c

  • Note 1: Equation of a line passing through the origin is y = mx, where m is the slope of the line.
  • Note 2: If the line is parallel to X-axis, then m = 0, therefore, the equation of line parallel to X-axis is y = c.

Point-Slope form of a Line

The equation of a line which passes through the point (x1 , y1) and has the slope m is y – y1 = m (x – x1)

Two Points form of a Line

The equation of a line passing through two points (x1, y1) and (x2, y2) is

$$\text{y -y}_{1}=\bigg(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\bigg)(x - x_{1})$$

Intercept form of a Line

The equation of a line which cuts off intercepts a and b respectively from X and Y-axis is

$$\frac{x}{a} +\frac{x}{b} = 1$$

Normal form or Perpendicular form
of a Line

The equation of the straight line upon which the length of the perpendicular from the origin is P and this perpendicular makes an angle a (Alpha) with

X-axis is x cos α + y sin α = P

Point Of Intersection Of Two Lines

Let the equation of two lines be

a1 x + b1 y + c1 = 0 ...(i)

a2 x + b2 y + c2 = 0 ...(ii)

Hence, coordinate of the point of intersection of equations (i) and (ii) are

$$\bigg(\frac{b_{1}c_{2}-b_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}},\frac{c_{1}a_{2}-c_{2}a_{1}}{a_{1}b_{2} - a_{2}b_{1}}\bigg)$$

Condition Of Concurreny Of Three Lines

Three lines L1  ≡ a1x + b1 y + c1 = 0, L2 ≡ a2x + b2y + c2 = 0 and L3 ≡ a3 x + b3 y + c3 = 0 are concurrent if

$$\begin{vmatrix}a_{1} &b_{1} &c_{1}\\a_{2} &b_{2} &c_{2}\\ a_{3} &b_{3} &c_{3}\end{vmatrix} = 0$$

Line Parallel To A Given Line

The equation of a line parallel to a given line ax + by + c = 0 is ax + by + λ = 0; where λ is a constant.

Line Perpendicular To A Given Line

The equation of a line perpendicular to a given line ax + by + c = 0 is bx – ay + λ = 0, where λ is a constant.

Angle Between Two Straight Lines When Their Equations Are Given

The acute angle θ between the lines a1x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0 is given by

$$\text{tan}\space\theta =\begin{vmatrix}\frac{a_{2}b_{1} - a_{1}b_{2}}{a_{1}a_{2}-b_{1}b_{2}}\end{vmatrix}$$

Distance Of A Point From A Line

The length of the perpendicular from a point (x1, y1) to a line ax + by + c = 0 is

$$\begin{vmatrix}\frac{a{x}_{1} + by_{1} + c}{\sqrt{a^{2} + b^{2}}}\end{vmatrix}$$

Distance Bewteen Two Parallel Lines

The distance between two parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is given by

$$\frac{|c_1-c_2|}{\sqrt{a^{2} +b^{2}}}$$

The length of perpendicular from the origin to a line

$$\text{ax + by + c = 0}\space\text{is}\space\frac{|\text{C}|}{\sqrt{a^{2} +b^{2}}}$$