Question

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

• A. $x^2-16xy-11y^2-12x+6y+21=0$
• B. $3x^2+16x+15y^2-4x-14y-1=0$
• C. $x^2+16x+11y^2-12x-6y+21=0$
• D. $3x^2+16x+15y^2-4x-14y+1=0$

Answer 1

The correct option is A $x^2-16xy-11y^2-12x+6y+21=0$

As we know that $P{S}^2=e^{2}P{M}^2$
$(x-2{)}^2+(y-1{)}^2=4\left[\frac{(x+2y-1{)}^2}{5}\right]$
$\Rightarrow 5[x^2+y^2-4x-2y+5]=4[x^2+4y^2+1+4xy-2x-4y]$
$\Rightarrow x^2-11y^2-16xy-12x+6y+21=0$