Question

Question 14 Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line 3x – 4y – 16 = 0.

Class X1 - Maths -Straight Lines Page 228

Answer 1

see attachment,
equation of line AB is 3x - 4y - 16 = 0
so, slope of line AB = 3/4

Let slope of line PQ = m
here, line PQ ⊥ line AB
so, slope of line PQ × slope of line AB = -1
m × (3/4) = -1
m = -4/3

now, equation of line PQ by using
y - y₁ = m(x - x₁)
here, (x₁, y₁) Ξ ( -1, 3)
y - 3 = -4/3(x + 1)
3(y - 3) + 4(x + 1) = 0
3y - 9 + 4x + 4 = 0
4x + 3y - 5 = 0

now, solve the equations 3x - 4y - 16 = 0 and 4x + 3y - 5 = 0 . we get co-ordinate of foot of perpendicular .
(3x - 4y - 16) × 3 + (4x + 3y -5) × 4 = 0
(9 + 16)x - 48 - 20 = 0
x = 68/25
y = 3 × 68/25 - 16 = -49/25

hence, ( 68/25, -49/25)