Let a line passing through the point ( 4 , 1 , 0 ) intersect the line L 1 ; 2 x − 1 = 3 y − 2 = 4 z − 3 at the point A ( α , β , γ ) and the line L 2 : x − 6 = y = − z + 4 at the point B ( a

Question

Let a line passing through the point ( 4 , 1 , 0 ) intersect the line L 1 ​ ; 2 x − 1 ​ = 3 y − 2 ​ = 4 z − 3 ​ at the point A ( α , β , γ ) and the line L 2 ​ : x − 6 = y = − z + 4 at the point B ( a , b , c ) . Then ​ 1 α a ​ 0 β b ​ 1 γ c ​ ​ is equal to

TestHub · Accepted Answer

​ L 1 ​ = 2 x − 1 ​ = 3 y − 2 ​ = 4 z − 3 ​ = p L 2 ​ = 1 x − 6 ​ = 1 y ​ = − 1 z − 4 ​ = q A ( 2 p + 1 , 3 p + 2 , 4 p + 3 ) B ( q + 6 , q , 4 − q ) D.R. of P A = 2 p − 3 , 3 P + 1 , 4 p + 3 D.R. of PB = q + 2 , q − 1 , 4 − q q + 2 2 p − 3 ​ = q − 1 3 p + 1 ​ = 4 − q 4 p + 3 ​ ​ ​ 2 pq − 2 p − 3 q + 3 = 3 pq + 6 p + q + 2 pq + rp + 4 q − 1 = 0 12 p − 3 pq + 4 − q = 4 pq + 3 q − 4 p − 3 7 pq − 16 p + 4 q − 7 = 0 8 p − 2 pq − 12 + 3 q = 4 pq + 8 p + 3 q + 6 6 pq = − 18 ∴ pq = − 3 8 p + 4 q = 4 ⇒ 2 p + q = 1 − 21 − 16 p + 4 q − 7 ⇒ 4 p − q = − 7 16 p − 4 q = − 28 ∴ p = − 1 , q = 3 A ( − 1 , − 1 , − 1 ) B ( 9 , 3 , 1 ) ​ ​ 1 − 1 9 ​ 0 − 1 3 ​ 1 − 1 1 ​ ​ = ​ 0 − 1 9 ​ − 1 − 1 3 ​ 0 − 1 1 ​ ​ = 1 ( − 1 + 9 ) = 8

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