Let `A(x_(1),y_(1))` and `B(x_(2),y_(2))` be two points on the parabola `y^(2) = 4ax`. If the circle with chord AB as a dimater touches the parabola, then `|y_(1)-y_(2)|` is equal to

Options

A. 4a
B. 8a
C. `6sqrt2a`
D. none of these

Answer

B

Explanation

Let the coordinates of A and B `(at_(1)^(2), 2at_(2))" and " (at_(2)^(2), 2at_(2))` respectively. Then, equation of the circle described on AB as a diameter is `(x-at_(1)^(2))(x-at_(2)^(2))+(y-2at_(1))(y-2at_(2))=0`
Suppose this cuts `y^(2)-4ax" at "(at^(2), 2at)`. Then,
`(at^(2)-at_(1)^(2))(at^(2)-at_(2)^(2))+(2at+2at_(1))(2at-2at_(2))=0`
`rArr" "(t+t_(1))(t+t_(2))+4=0`
`rArr" "t^(2)+t(t_(1)+t_(2))+t_(1)t_(2)+4=0`
If the circle touches the parabola, then this equation must give equal values of t.
`:." "(t_(1)+t_(2))^(2)-4(t_(1)t_(2)+4)=0`
`rArr" "(t_(1)-t_(2))^(2)=16rArr|t_(1)-t_(2)|=4rArr|y_(1)-y_(2)|=8a`