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Question:

A line passing through the origin is perpendicular to the lines l1:(3+t)^i + (a+2t)^j + (4+2t)^k, −∞<t<∞, l2:(3+2s)^i + (3+2s)^j + (2+s)^k, −∞<t<∞. 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, a,0), (1,1,1), (73, 73, 53), (79, 79, 89)

(73, 73, 53)

(a, a,0)

(1,1,1)

(79, 79, 89)

Solution:

The common perpendicular is along ∣∣∣∣∣^i^j^k122221∣∣∣∣∣=a^i+3^j+a^kLet M=(2λ,aλ,2λ)So, 2λ−a=aλ+12=2λ−a=>λ=1So,M=(2,a,2)Let the required point be PGiven that PM=√17=>(3+2s−a)2+(3+2s+3)2+(2+s−a)2=17=>9s2+28s+20=0=>s=−2, −109So,P=(a,a,0)or(79,79,89).