Question

If co-ordinates of points A, B, C be (-1, 5), (0, 0) and (2, 2) respectively and D be the middle point of BC, then the equation of perpendicular from B to the line AD is
(a) x + 2y = 0
(b) 2x + y = 0
(c) x - 2y = 0
(d) 2x – y = 0

Answer 1

Hint: Find slope of AD and then use point-slope formula for line equation.
Line equation passing through (a, b) with slope m is:
(y - b) = m.(x – a)

Complete Step-by-Step solution:
First we need to find point D(x, y).
Given, D is the midpoint of BC.
Co-ordinates of D will be the average of coordinates of B, C.
So by calculating x-co-ordinate of D, we get:
x = average of (0, 2)
\[x=\dfrac{0+2}{2}\]
By simplifying, we get:
x = 1…..(1)
So by calculating y-coordinate of D, we get:
y = average of (0, 2)
\[y=\dfrac{0+2}{2}\]
y = 1…..(2)
By equation (1) and equation (2), we get:
D = (1, 1).
Given, A = (-1, 5).
We know that:
Let slope of a line passing through two points (a, b) and (c, d) is m, then:
\[m=\dfrac{d-b}{c-a}.....\left( 3 \right)\]
So by applying above condition to A and D, the values are:
a = 1, b = 1, c = -1, d = 5.
Le slope of AD be m.
Substituting values of a, b, c, d into equation (3), we get:
\[m=\dfrac{5-1}{-1-1}=\dfrac{4}{-2}=-2\]
By above equation, we get:
m = -2…..(4)
We need an equation perpendicular to AD.

Let the slope of the required line be n.
if two straight lines with slope a, b are perpendicular then
\[a\times b=-1\].
By applying above condition, we get:
\[m\times n=-1\]
\[n=-\dfrac{1}{m}.....\left( 5 \right)\]
By substituting equation (5) in equation (4), we get:
\[n=\dfrac{1}{2}.....\left( 6 \right)\]
So now we need a line equation of slope n and passing through B (0, 0). Now we need to use a point-slope formula to find an equation. Line equation passing through (a, b) with slope m is:
(y - b) = m.(x – a)…..(7)
So from equation (6) and B (0,0)
\[m=\dfrac{1}{2}\]
b = 0
a = 0
By substituting values of m, b, a in equation (7), we get:
\[\left( y-0 \right)=\dfrac{1}{2}\left( x-0 \right)\]
By simplifying, we get:
2y = x
2y – x = 0
By multiplying with -1 on both sides, we get:
x – 2y = 0.
\[\therefore \]The line equation from B which is perpendicular to AD is x - 2y = 0.
So option (c) is correct.

Note: Alternate method is to find the line equation of AD and find a foot of perpendicular from B to AD, then find an equation passing through B and the foot of perpendicular.