Question

The point P is the foot of perpendicular from A (-5, 7) to the line 2x - 3y + 18 = 0.

Determine :

(i) the equation of the line AP

(ii) the co-ordinates of P

Answer 1

(i) The given equation is

2x - 3y + 18 = 0

3y = 2x + 18

y = x + 6

Slope of this line =

Slope of a line perpendicular to this line =

(x₁, y₁) = (-5, 7)

The required equation of the line AP is given by

y - y₁ = m(x - x₁)

y - 7 = (x + 5)

2y - 14 = -3x - 15

3x + 2y + 1 = 0

(ii) P is the foot of the perpendicular from point A.

So P is the point of intersection of the lines 2x - 3y + 18 = 0 and 3x + 2y + 1 = 0.

2x - 3y + 18 = 0 $\Rightarrow$ 4x - 6y + 36 = 0

3x + 2y + 1 = 0 $\Rightarrow$ 9x + 6y + 3 = 0

Adding the two equations, we get,

13x + 39 = 0

x = -3

3y = 2x + 18 = -6 + 18 = 12

y = 4

Thus, the coordinates of the point P are (-3, 4).