A circle is drawn having centre at `C (0,2)` and passing through focus (S) of the parabola `y^2=8x`, if radius (CS) intersects the parabola at point P, then

Options

A. distance of point P from directrix is `(8-4sqrt(2))`
B. distance of point C from point P is `(6sqrt(2)-8)`
C. angle subtended by intercept made by circle on directrix at its centre is `(pi)/(2)`
D. point P is the midpoint of C and S

Answer

A, B, C

Explanation

1,2,3
Image of S w.r.t. point C(0,2) is Q(-2,4). Thus, SQ is diameter. So circle must pass through R (foot of directrix on axis of parabola)

Equation of CS is x+y=2 and parabola is `y^(2)=8x`.
Solving, we get x-coordinate of P as `6-4sqrt(2)`.
`:.PM=8-4sqrt(2)`
`CP=CS-SP=2sqrt(2)-(8-4sqrt(2))=6sqrt(2)-8`
Slope of `CQxx` Slope of CR=-1