A line `ax +by +c = 0` through the point `A(-2,0)` intersects the curve `y^(2)=4a` in P and Q such that `(1)/(AP) +(1)/(AQ) =(1)/(4)` (P,Q are in 1st quadrant). The value of `sqrt(a^(2)+b^(2)+c^(2))` is

Options

A. 2
B. 4
C. 6
D. 8

Answer

B

Explanation

Let `P(-2+r cos theta, r sin theta)`.
`rArr r^(2) sin^(2) theta - 4r cos theta +8 =0`
`rArr (1)/(AP) + (1)/(AQ) =(1)/(r_(1)) +(1)/(r_(2)) =(1)/(4)`
`rArr cos theta =(1)/(2) rArr tan theta = sqrt(3)`
So, equation of line is
`y -0 = sqrt(3) (x+2)`
`rArr sqrt(3) x -y +2sqrt(3) =0`
So `sqrt(a^(2)+b^(2)+c^(2)) =4`.