Class 10: Equation of a Line – Lecture Notes – ICSE / ISC / CBSE Mathematics Portal for K12 Students

Basic Concept of a Line

Any point, that satisfies the equation of a line, will lie on the line.
Also, any point through which a line passes, will always satisfy the equation of that line.
Every straight line can be represented by a linear equation in two dimensions ( $x \ and \ y$ plane).

Inclination of a Line

The angle $\theta$ that the line makes with the $x-axis$ is called the “inclination of the line”. The angle $\theta$ is when measured in anti-clockwise direction the inclination is positive and if measured clock-wise then the inclination is negative.

Figure 1

Figure 2

Note:

Inclination angle $\theta$ for $x-axis$ is $0^o$ . Also all lines parallel to $x-axis$ will also have the inclination angle as $0^o$ .
Inclination angle $\theta$ for $y-axis$ is $90^o$ . Also all lines parallel to $y-axis$ will also have the inclination angle as $90^o$ .

Slope (or Gradient) of a Line

The slope of the line is the tangent of its inclination and is denoted by $m$ i.e. Slope $m = \tan \theta$

Therefore

Slope of the $x-axis = \tan 0^o= 0$

Slope of the $y-axis = \tan 90^o = \infty$ (not defined)

The slope of the line is positive if the line makes an acute angle with $x-axis$ when measured in anti-clock wise direction i.e. $\tan \theta$ is positive.

The slope of the line is negative if the line makes and obtuse angle with $x-axis$ when measured in anti-clockwise direction. i.e. $\tan \theta$ is negative.

Formula for “Slope of Line passing through two points”

Let the two points be $(x_1, y_1)$ and $(x_2, y_2)$

$\displaystyle m = \tan \theta = \frac{y_2-y_1}{x_2-x_1} = \frac{y_1-y_2}{x_1-x_2}$

Parallel Lines

Two lines having inclination as $\theta \ and \ \alpha$ will be parallel if

$\alpha=\theta \ or \ \tan \alpha=tan\theta$

i.e. slope of the two lines is equal and hence they are parallel lines.

Perpendicular Lines

Two lines having inclination as $\theta$ and $\alpha$ will be perpendicular if:

$\theta = 90^o+\alpha$

$\Rightarrow \tan \theta=tan \ (90^o+\alpha)$

$\Rightarrow \tan \theta=- \cot \alpha$

$\displaystyle \Rightarrow \tan \theta=- \frac{1}{\tan \alpha}$

$\Rightarrow \tan \theta \times \tan \alpha = -1$

or $m_1 \times m_2 = -1$ where $m_1 \ and \ m_2$ are slopes of the two lines respectively.

Note:

If two lines are perpendicular, then the product of their slopes is $-1$ . Conversely, if the product of the two lines is $-1$ , they are perpendicular.
Example: If the slope of a line is $2$ , then the slope of another line parallel to this line would be $2$ . Also the slope of a line perpendicular to this line would be $- \frac{1}{2}$ .
A line whose slope is $m=0$ , is parallel to $x-axis$ .
Also the slope of $y-axis$ is not defined, all likes parallel to $y-axis$ will have slope which is not defined.