If and k are segments of a focal chord of the parabola `y^(2)= 4ax`, then k =

Options

A. ` ( ab ) /( a- b ) `
B. ` ( a ) /( b - a ) `
C. ` ( b ) /( b - a ) `
D. ` ( ab ) /( b - a ) `

Answer

D

Explanation

For latusrectum PQ,
` y ( t _ 1 + t _ 2 ) - 2x - 2a t _ 1 t _ 2 = 0 `
and ` t _ 1 t _ 2 = - 1 `
For any point ` P ( x , y ) `, then focal distance is ` a + x `.
` therefore b = a + x = a + a t _ 1 ^ 2 = a ( 1 + t _ 1 ^ 2 ) ` ...(i )
` c = a + x = a + at _ 2 ^ 2 = a + ( a ) /( t _ 1 ^ 2 ) ( because t _ 1 t _ 2 = - 1 ) `
` = ( a ( 1 + t _ 1 ^ 2 ) ) /( t _ 1^ 2) " " `... (ii)
` therefore (b )/ (c ) = t _ 1 ^ 2 `
` therefore ` From Eq. (i),
` b = a ( 1 + (b)/ (c )) `
` rArr b = a + ( ab ) / ( c ) `
` therefore c = ( ab ) /( b - a ) `