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