3D Geometry 3 Question 52

52. In R3, let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes P1:x+2yz+1=0 and P2:2xy+z1=0. Let M be the locus of the foot of the perpendiculars drawn from the points on L to the plane P1. Which of the following point(s) lie(s) on M ?

(2015 Adv.)

(a) 0,56,23

(b) 16,13,16

(c) 56,0,16

(d) 13,0,23

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Answer:

Correct Answer: 52. (b, d)

Solution:

  1. Since, L is at constant distance from two planes P1 and P2. Therefore, L is parallel to the line through intersection of P1 and P2.

 DR’s of L=|i^j^k^121211|=i^(21)j^(1+2)+k^(14)=i^3j^5k^

DR’s of L are (1,3,5) passing through (0,0,0).

Now, equation of L is

x01=y03=z05

For any point on L,x1=y3=z5=λ

[say]

i.e. P(λ,3λ,5λ)

If (α,β,γ) is foot of perpendicular from P on P1, then

αλ1=β+3λ2=γ+5λ1=kα=λ+k,β=2k3λ,γ=k5λ which satisfy P1:x+2yz+1=0(λ+k)+2(2k3λ)(k5λ)+1=0k=16x=16+λ,y=133λ,z=165λ

which satisfy options (a) and (b).



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