If a line with direction ratios 2 : 2 : 1 intersects the lines 3 x − 7 = 2 y − 5 = 1 z − 3 and 2 x − 1 = 4 y + 1 = 3 z + 1 at A and B then AB =

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If a line with direction ratios 2 : 2 : 1 intersects the lines 3 x − 7 ​ = 2 y − 5 ​ = 1 z − 3 ​ and 2 x − 1 ​ = 4 y + 1 ​ = 3 z + 1 ​ at A and B then AB =

TestHub · Accepted Answer

Let A ( 7 + 3 α , 5 + 2 α , 3 + α ) and B ( 1 + 2 β , − 1 + 4 β , − 1 + 3 β ) . The direction ratios (D.R.'s) of A B are in the ratio 2 : 2 : 1 . Therefore, we can write the following proportion: 2 6 + 3 α − 2 β ​ = 1 3 + α − 2 β ​ = 1 4 + α − 3 β ​ Solving these equations yields α = − 2 and β = 1 . Substituting these values back into the coordinates of A and B , we get: A ( 1 , 1 , 1 ) and B ( 3 , 3 , 2 ) . Hence AB=3

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