If the square of the shortest distance between the lines 1 x − 2 = 2 y − 1 = − 3 z + 3 and 2 x + 1 = 4 y + 3 = − 5 z + 5 is n m , where m , n are coprime numbers, then m + n is equal to

Question

If the square of the shortest distance between the lines 1 x − 2 ​ = 2 y − 1 ​ = − 3 z + 3 ​ and 2 x + 1 ​ = 4 y + 3 ​ = − 5 z + 5 ​ is n m ​ , where m , n are coprime numbers, then m + n is equal to :

TestHub · Accepted Answer

​ a = ( 2 , 1 , − 3 ) b = ( − 1 , − 3 , − 5 ) p ​ × q ​ = ​ i ^ 1 2 ​ j ^ ​ 2 4 ​ k ^ − 3 − 5 ​ ​ = 2 i ^ − j ^ ​ ​ ​ b − a = − 3 i ^ − 4 j ^ ​ − 2 k ^ S d ​ = ∣ p ​ × q ​ ∣ ∣ ( b − a ) ⋅ ( p ​ × q ​ ) ∣ ​ − = 5 ​ 2 ​ ( S d ​ ) 2 = 5 4 ​ m = 4 , n = 5 ⇒ m + n = 9 ​

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