Question

The equation of the hyperbola whose directrix is $x+2 y=1$, focus $(2,1)$ and eccentricity 2 will be

• A. $x^2-16 x y-11 y^2-12 x+6 y+21=0$
• B. $3 x^2+16 x y+15 y^2-4 x-14 y-1=0$
• C. $3 x^2+16 x y+11 y^2-12 x-6 y+21=0$
• D. None of these

Answer 1

Answer: (A)

$PS = ePM$
$\Rightarrow PS ^2= e ^2 PM ^2$
$\Rightarrow(x-2)^2+(y-1)^2=4 \frac{(x+2 y-1)^2}{5}$
$\Rightarrow x^2+y^2-4 x-2 y+5=\frac{4}{5}\left(x^2+4 y^2+1+4 x y-4 y-2 x\right)$
$\Rightarrow 5 x^2+5 y^2-20 x-10 y+25=4 x^2+16 y^2+4+16 x y-16 y-8 x$
$x^2-16 x y-11 y^2-12 x+6 y+21=0$