If a line `y=3x+1` cuts the parabola `x^2-4x-4y+20=0` at `A and B ,` then the tangent of the angle subtended by line segment `A B` at the origin is (a) `(8sqrt(3))/(205)` (b) `(8sqrt(3))/(209)` (c)`(8sqrt(3))/(215)` (d) none of these

Options

A. `8sqrt(3)//205`
B. `8sqrt(3)//209`
C. `8sqrt(3)//215`
D. none of these

Answer

B

Explanation

(2) The joint equation of OA and OB is
`x^(2)-4x(y-3x)-4y(y-3x)+20(y-3x)^(2)=0`
(Making the equation of the parabola homogeneous using straight line)
`or193x^(2)+16y^(2)-112xy=0`
`tantheta=(2sqrt(h^(2)-ab))/(a+b)`
`=(2sqrt(56^(2)-193xx16))/(193+16)`
`=(8sqrt(3))/(209)`