Q. 165.0( 2 Votes )
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]
Answer :
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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