Let the point ( − 1 , α , β ) lie on the line of the shortest distance between the lines − 3 x + 2 = 4 y − 2 = 2 z − 5 and − 1 x + 2 = 2 y + 6 = 0 z − 1 . Then ( α − β ) 2 is equal to_____

Question

Let the point ( − 1 , α , β ) lie on the line of the shortest distance between the lines − 3 x + 2 ​ = 4 y − 2 ​ = 2 z − 5 ​ and − 1 x + 2 ​ = 2 y + 6 ​ = 0 z − 1 ​ . Then ( α − β ) 2 is equal to___________

TestHub · Accepted Answer

​ P ( − 3 λ − 2 , 4 λ + 2 , 2 λ + 5 ) Q ( − μ − 2 , 2 μ − 6 , 1 ) DRS of PQ = ( 3 λ − μ , 2 μ − 4 λ − 8 , − 2 λ − 4 ) DRS of PQ = ​ i ^ − 1 − 3 ​ j ^ ​ 2 4 ​ k ^ 0 2 ​ ​ = ( 4 i ^ + 2 j ^ ​ + 2 k ^ ) ​ OR ​ ( 2 , 1 , 1 ) 2 3 λ − μ ​ = 1 2 μ − 4 λ − 8 ​ = 1 − 2 λ − 4 ​ ⇒ μ = λ + 2&7 λ = μ − 8 λ = − 1 μ = 1 Q : ( − 3 , − 4 , 1 ) L PQ ​ = 2 x + 3 ​ = 1 y + 4 ​ = 1 z − 1 ​ ( − 1 , α , β ) ⇒ 1 = 1 α + 4 ​ = 1 β − 1 ​ ⇒ α = − 3 , β = 2 ( α − β ) 2 = 25 ​

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