Question

If, for a rectangular hyperbola, a focus is $(1,2)$ and the corresponding directrix is $x+y=1$ then the equation of the rectangular hyperbola is :

• A. $x^2-y^2=2$
• B. $xy-y+2=0$
• C. $xy+y-2=0$
• D. none of these

Answer 1

The correct option is C $xy+y-2=0$

Given focus $S(2,1)$ and directrix $x+y-1=0$
Let $M$ and $P$ are the foot of perpendicular and point on the conic.
and we know eccentricity of a rectangular hyperbola is $\sqrt{2}$
Hence using conic definition $\frac{P{S}^2}{P{M}^2}=e^2\Rightarrow P{S}^2=e^2.P{M}^2$
$\Rightarrow (x-1{)}^2+(y-2{)}^2=2.{\left|\frac{x+y-1}{\sqrt{2}}\right|}^2$
$\Rightarrow x^2+y^2-2x-4y+5=x^2+y^2=x^2+y^2+2xy-2x-2y+1$
$\Rightarrow xy+y-2=0$
Hence, option 'C' is correct.