Equation of Straight Lines in all Forms

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| = \(3\sqrt{2}\)

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