The point of intersection of the lines ` ( x + 1 )/( 3 ) = ( y + 3 ) /( 5 ) = ( z+ 5 ) /( 7 ) and ( x - 2 )/( 1) = ( y - 4 ) /( 3 ) = ( z - 6 )/ ( 5 ) ` is

Options

A. ` ( ( 1 )/ ( 2), ( 1 ) /(2), - ( 3 ) / ( 2)) `
B. ` ( - ( 1 ) / ( 2 ), - ( 1 ) / ( 2 ), ( 3 ) / ( 2 )) `
C. ` (( 1 )/ ( 2 ), - (1)/ ( 2 ) , - ( 3 ) / ( 2)) `
D. ` ( - ( 1 ) / ( 2) , ( 1 ) / ( 2 ), ( 3 ) / ( 2) ) `

Answer

C

Explanation

Let ` ( x + 1 ) /( 3 ) = ( y + 3 )/ ( 5 ) = ( z + 5 ) / ( 7 ) = lamda " "`…(i)
Any point on the line is ` ( 3lamda - 1 , 5lamda - 3, 7 lamda - 5 ) `
Again, let
` ( x - 2 ) / ( 1 ) = ( y- 4 ) / ( 3) = ( z- 6 ) / ( 5) = mu " " `...(ii)
Any point on the line is
` ( mu + 2, 3mu + 4, 5 mu + 6 ) `
For intersection , they have a common point
` therefore ( 3 lamda - 1 ) = ( mu + 2 ) `,
` ( 5lamda - 3 ) = ( 3 mu + 4 ) `,
and ` ( 7 lamda - 5 ) = ( 5mu + 6 ) `
From first two , we have
` mu = 3 lamda -3 " " `...(iii)
and ` 3 mu = 5lamda - 7 " " `...(iv)
From Eqs (iii) and (iv) ,we have
` 3 ( 3 lamda - 3 ) = 5 lamda - 7 rArr lamda = ( 1 ) / ( 2 ) `
Point of intersection is ` (( 3 ) / (2 ) - 1, (5 ) / ( 2 )- 3, ( 7 )/ ( 2 ) - 5 ) = (( 1) / ( 2 ) , - ( 1 ) / (2 ) , - ( 3 ) / (2 )) `