Question

A hyperbola has one focus at the origin and its eccentricity = $\sqrt{2}$ and one of its directrix is $x+y+1=0$. If the equation to its asymptotes is $x+m=0andy+m=0$.Find $m$.

Answer 1

Given: focus at (0,0)

eccentricity,$e=\sqrt{2}$

directric:$x+y+1=0$-------------(1)

Asymptotes: $x+m=0\text{and}y+m=0$----------------(2) & (3)

We know, Hyperbola : $(x+m)(y+m)+\lambda =0$----------------(4)

We know, $SP=ePM$

$\Rightarrow {(x-h)}^2+{(y-k)}^2+\lambda =e^2\frac{{(x+y+1)}^2}{1^2+1^2}$

Where (h,k) is focus,

and $e=\sqrt{2}$

$\Rightarrow {(x-0)}^2+{(y-0)}^2=2.\frac{{(x+y+1)}^2}{2}\Rightarrow x^2+y^2=(x^2+y^2+1+2xy+2y+2x)\Rightarrow 2xy+2y+2x+1=0\Rightarrow xy+x+y+\frac{1}{2}=0$

On comparing (5) and (4) we get coefficient of x & y

$\Rightarrow m=1$