Question

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 $AB$ at origin is

• A. $\frac{8\sqrt{3}}{205}$
• B. $\frac{8\sqrt{3}}{209}$
• C. $\frac{8\sqrt{3}}{215}$
• D. none of these

Answer 1

The correct option is B $\frac{8\sqrt{3}}{209}$

Joint equation of $OA$ and $OB$ is found by making equation of parabola homogeneous using straight line

$x^2-4x(y-3x)-4y(y-3x)+20(y-3x{)}^2=0$

$\Rightarrow x^2(1+12+180)-y^2(4-20)-xy(4-12+120)=0$

$\Rightarrow 193x^2+16y^2-112xy=0$

Here $a=193,b=16,2h=-112$

$\tan \theta =\frac{2\sqrt{h^2-ab}}{a+b}$

$=\frac{2\sqrt{{56}^2-193\times 16}}{193+16}=\frac{8\sqrt{3}}{209}$