# Find the coordinates of the foot of the perpendicular drawn from the point A( - 1, 8, 4) to the line joining the points B(0, - 1, 3) and C(2, - 3, - 1). Hence find the image of the point A in the line BC.[CBSE 2016]

The Cartesian equation of line passing through A(x,y,z),B(x1,y1,z1) and C(x2, y2, z2) is given by

B is (0, - 1,3) and C is (2, - 3, - 1)

Substituting the values we get

General coordinates of P is (2λ, - 2λ - 1, - 4λ + 3)

Direction ratio of AP(2λ + 1, - 2λ - 9, 4λ - 1)

AP is perpendicular to BC

2(2λ + 1) - 2( - 2λ - 9) - 4( - 4λ - 1) = 0

4λ + 2 + 4λ + 18 + 16λ + 4 = 0

24 + 24λ = 0

λ = - 1

P( - 2,1,7)

Coordinates of Foot of perpendicular are ( - 2,1,7)

Coordinates of image of A is P’(a,b,c) is

a = - 3

b = - 6

c = 10

P’( - 3, - 6,10)

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