Question

The line segment joining the foci of the hyperbola $x^2-y^2+1=0$ is one of the diameters of a circle. The equation of the circle is :

• A. $x^2+y^2=4$
• B. $x^2+y^2=\sqrt{2}$
• C. $x^2+y^2=2$
• D. $x^2+y^2=2\sqrt{2}$

Answer 1

The correct option is C $x^2+y^2=2$

$x^2-y^2+1=0⟹(x-0{)}^2+(y-0{)}^2=0$

Foci of the given hyperbola $=(0,\pm \sqrt{2})$

Centre of the hyperbola is $(0,0)$

Diameter of the circle is the distance between foci of the hyperbola and which is given by

$D=\sqrt{(0-0{)}^2+(\sqrt{2}+\sqrt{2}{)}^2}=2\sqrt{2}$

Centre of the circle will be same as that of the hyperbola

$\therefore Centre=(0,0)$ and $Radius=\sqrt{2}$
$\therefore$ Equation of the circle is $x^2+y^2=2$