Q16 of 34 Page 11

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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