बिंदु `(1,2,-4)` से जाने वाली और दोनों रेखाओ `(x-8)/(3)=(y+19)/(-16)=(z-10)/(7) ` और`(x-15)/(3)=(y-29)/(8)=(z-5)/(-5)`पर लंब रेखा का सदिश समीकरण ज्ञात कीजिए l

Options

A. ` r = ( hati + 2 hatj - 4 hatk ) + lamda ( 2 hati + 3 hat j + 6 hatk ) `
B. ` r = ( 2 hati + 3 hatj - 6 hatk ) + lamda ( hati + 2hatj - 4hatk ) `
C. ` r = ( hati + 2 hatj - 4hatk ) + lamda ( 3hati + 8 hatj - 5hatk ) `
D. ` r = ( hati + 2hatj - 4hatk ) + lamda ( 3 hati - 16 hatj - 7 hatk ) `

Answer

A

Explanation

Any line through ` ( 1, 2, - 4) ` can be written as
` ( x - 1 ) / ( a ) = ( y -2 ) / ( b ) = ( z+ 4 ) / ( c) " " `…(i)
where a , b, c are the direction ratios of line (i).
Now the line (i) be perpendicular to the lines
` ( x- 8 ) /(3 ) = ( y + 19) / ( -16 ) =( z - 10 ) /( 7 ) `
and ` ( x - 15 ) / ( 3 ) = ( y - 29) / ( 8 ) = ( z - 5 ) / ( - 5 ) `
` therefore 3a - 16 b + 7 c = 0 " " `...(ii)
and ` 3a +8 b - 5 c = 0 " " `...(iii)
By cross - multiplication , we have
` ( a ) / ( 80 - 56 ) = ( b ) /( 21 + 15 ) = ( c ) /( 24 - 48 ) `
` rArr ( a ) /( 24 ) = ( b ) /( 36 ) = ( c ) /( 72) `
` rArr (a )/ ( 2 ) = ( b ) / ( 3 ) = (c ) / ( 6) = lamda " " ( say) `
` therefore a = 2 lamda , b = 3lamda and c = 6lamda `
The equation of required line which passes through the point ` ( 1 , 2, -4 ) ` and perpendicular to vector ` 2 hati + 3 hatj + 6hatk ` is ` r = ( hati + 2 hatj - 4hatk ) + lamda ( 2 hati + 3 hatj + 6 hatk ) `