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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


Solution

(i) The given equation is

2x - 3y + 18 = 0

3y = 2x + 18

y = 2 over 3x + 6

Slope of this line = 2 over 3

Slope of a line perpendicular to this line = fraction numerator begin display style fraction numerator negative 1 over denominator 2 end fraction end style over denominator 3 end fraction equals fraction numerator negative 3 over denominator 2 end fraction

(x1, y1) = (-5, 7)

The required equation of the line AP is given by

y - y1 = m(x - x1)

y - 7 = fraction numerator negative 3 over denominator 2 end fraction(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  4x - 6y + 36 = 0

3x + 2y + 1 = 0   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).


Mathematics
Concise Mathematics
Standard X

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