The reflection of the point P ( 1 , 0 , 0 ) in the line 2 x − 1 = − 3 y + 1 = 8 z + 10 is

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The reflection of the point P ( 1 , 0 , 0 ) in the line 2 x − 1 ​ = − 3 y + 1 ​ = 8 z + 10 ​ is

TestHub · Accepted Answer

Any point Q on the line is ( 2 r + 1 , − 3 r − 1 , 8 r − 10 ) . The direction ratios of PQ are 2 r , − 3 r − 1 , 8 r − 10 . PQ is perpendicular to the line if 2 ( 2 r ) − 3 ( − 3 r − 1 ) + 8 ( 8 r − 10 ) = 0 , i.e. 77 r − 77 = 0 , so r = 1 and the foot of the perpendicular is Q ( 3 , − 4 , − 2 ) . Let R ( a , b , c ) be the reflection, so Q is the mid-point of PR: 2 a + 1 ​ = 3 , 2 b ​ = − 4 , 2 c ​ = − 2 . So R = ( 5 , − 8 , − 4 ) .

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