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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

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Solution

(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 ⇒ 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).

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