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.

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   $(x_2, y_2)$

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

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:

$\theta = 90^o+\alpha$

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

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

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

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

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.

Condition for Collinearity of Three Points

Let there be three points $A(x_1, y_1), B(x_2, y_2) \ and \ C(x_3, y_3)$

If the slope of segment AB and BC is the same, then they are all on the same line and are collinear.

$m = \frac{y_2-y_1}{x_2-x_1} = \frac{y_3-y_2}{x_3-x_2}$

X-intercepts & Y-intercepts

Refer to first figure.

If a line meets the $x-axis$, then the distance from origin to the point of intercept is called $x-intercept$.

Similarly, if a line meets the  $y-axis$, then the distance from origin to the point of intercept is called  $y-intercept$.

Equation of a Line

Slope-Intercept Form: When the slope $(m)$ is given and the intercept on the $y-axis$ is given.

The the equation is:

$y = mx+c$ where $m$ is the slope and $c = intercept \ on \ y-axis$.

Point-Slope form: When the slope of the line and a point $(x_1, y_1)$ on a line is given

The equation is:

$y-y_1 = m(x-x_1)$

Equally Inclined Lines

This means that the lines make equal angle from both the coordinate axes. As shown in the figure, lines $AB \ and \ CD$ are equally inclined.

For $AB: Inclination \ \theta=45^o$. Therefore slope $= tan(45^o) = 1$

For $CD: Inclination \ \theta=-45^o$. Therefore the slope $=tan(-45^o) = -1$