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Question

A line l passing through the origin is perpendicular to the lines
l1:(3+t)^i+(1+2t)^j+(4+2t)^k, <t<
l2:(3+2s)^i+(3+2s)^j+(2+s)^k, <s<

Then, the coordinate(s) of the point(s) on l2 at a distance of 17 from the point of intersection of l and l1 is (are)

A
(73,73,53)
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B
(1,1,0)
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C
(1,1,1)
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D
(79,79,89)
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Solution

The correct option is D (79,79,89)
l1:(3^i^j+4^k)+t(^i+2^j+2^k)
l2:(3^i+3^j+2^k)+s(2^i+2^j+^k)

Direction ratios of line l is
∣ ∣ ∣^i^j^k122221∣ ∣ ∣=2^i+3^j2^k

So, equation of line l is
x2=y3=z2=λ
Any point on line l is (2λ,3λ,2λ)

Intersection point of l and l1:
2λ=3+t (1)
3λ=1+2t (2)
2λ=4+2t
Solving equations (1) and (2), we get
λ=1
Point of intersection is (2,3,2)

So, (3+2s2)2+(3+2s+3)2+(2+s2)2=17
9s2+28s+20=0
s=2,109

Required points are (1,1,0) and (79,79,89)

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