Here, you will learn equation of straight lines in all forms i.e. slope form, intercept form, normal form and parametric form etc.

Let’s begin –

A relation between x and y which is satisfied by co-ordinates of every point lying on a line is called equation of the straight lines. Here, remember that every one degree equation in variable x and y always represent a straight line i.e. ax + by + c = 0 ; a & b \(\ne\) 0 simultaneously.

(a) Equation of a line parallel to x-axis at a distance ‘a’ is y = a or y = -a.

(b) Equation of x-axis is y = 0

(c) Equation of a line parallel to y-axis at a distance ‘b’ is x = b or x = -b.

(d) Equation of y-axis is x = 0.

## Different Equation of Straight Lines

**(a) Slope Intercept form :**

Let m be the slope of a line and c its intercept on y-axis. Then the equation of this straight line is written as

y = mx + c.

**(b) Point Slope form :**

Let m be the slope of a line and it passes through a point (\(x_1,y_1\)), then its equation is written as :

y – \(y_1\) = m(x – \(x_1\)).

**(c) Two Point form :**

Equation of a line passing through two points (\(x_1,y_1\)) and (\(x_2,y_2\)) is written as

y – \(y_1\) = \(y_2-y_1\over {x_2-x_1}\)(x – \(x_1\)).

**(d) Intercept form :**

If a and b are the intercepts made by a line on the axes of x and y, its equation is written as :

\(x\over a\) + \(y\over b\) = 1

Length of intercept of line between the coordinate axes = \(\sqrt{a^2+b^2}\)

**(e) Normal form :**

If p is the length of perpendicular on a line from the origin, and \(\alpha\) the angle which this perpendicular makes with positive x-axis, then the equation of this line is written as

xcos\(\alpha\) + ysin\(\alpha\) = p (p is always positive) where 0 \(\le\) \(\alpha\) < 2\(\pi\).

**(f) Parametric form :**

To find the equation of a straight line which passes through a given point A(h,k) and makes a given angle \(\theta\) with the positive direction of the axis. P(x,y) is any point on the line.

Let AP = r, then x – h = rcos\(\theta\), y – k = rsin\(\theta\) & \(x – h\over {cos\theta}\) = \(y – k\over {sin\theta}\) = r is the equation of straight line.

**(g) General form :**

We know that a first degree equation in x and y, ax + by + c = 0 always represent a straight line. This form is known as general form of straight line.

(i) Slope of this line = -\(a\over b\)

(ii) Intercept by this line on x-axis = -\(c\over a\) and Intercept by this line on y-axis = -\(c\over b\)

(iii) To change the general form of a line to normal form, first take c to right hand side and make it positive, then divide the whole equation by \(\sqrt{a^2+b^2}\).

Example : Equation of a line which passes through point A(2,3) and makes an angle of 45 with x-axis. If this line meet the line x + y + 1 = 0 at a point P then distance AP is-

Solution : Here \(x_1\) = 2, \(y_1\) = 3 and \(\theta\) = 45

Hence \(x-2\over cos45\) = \(y-3\over sin45\) = r

from first two parts \(\implies\) x – 2 = y – 3 \(\implies\) x – y + 1 = 0

Co-ordinate of point P on this line is (2+\(r\over \sqrt{2}\), 3+\(r\over \sqrt{2}\)).

If this point is on line x + y + 1 = 0 then

(2+\(r\over \sqrt{2}\)) + (3+\(r\over \sqrt{2}\)) + 1 = 0 \(\implies\) r = -\(3\sqrt{2}\) ; |r| = \(3\sqrt{2}\)