The radius of the circle whose centre is (-4,0) and which cuts the parabola `y^(2)=8x` at A and B such that the common chord AB subtends a right angle at the vertex of the parabola is equal to

Options

A. 4
B. 3
C. `sqrt(18)`
D. 5

Answer

A

Explanation

Let be the radius of the circle. Then, its equation is
`(x+)^(2)+y^(2)=r^(2)" ...(i)"`
This cute the parabola `y^(2)=8x` at points `A(x_(1), y_(1))` and `B(x_(2), y_(2))`.
The abcissae of A and B are the roots of the equation
`(x+4)^(2)+8x=r^(2)or, x^(2)+16x+16-r^(2)=0`
`:." "x_(1)x_(2)=16-r^(2)" ....(ii)"`
The ordinates of A and B are given by `y_(1)^(2)=8x_(1)" and "y_(2)^(2)=8x_(2)` respectively.
`:." "y_(1)=+-2sqrt2x_(1)" and "y_(2)=+-2sqrt2x_(2)`
Since AB subtends a right angle at the vertex of the parabola.
`:." "y_(1)/x_(1)xxy_(2)/x_(2)=-1`
`rArr" "x_(1)x_(2)+y_(1)y_(2)=0`
`rArr" :x_(1)x_(2)+8x_(1)x_(2)=0`
`rArr" "x_(1)x_(2)=0" "["Using (i)"]`
`rArr" "16-r^(2)=0rArrr=4`