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 $x^2+y^2=2$

$x^2-y^2+1=0$
$\Rightarrow y^2-x^2=1$
This is a rectangular hyperbola whose transverse axis is y axis.
$c=\sqrt{2}$
Therefore, coordinates of foci are $(0,\sqrt{2}),(0,-\sqrt{2})$
So, equation of the circle with focii as the end points of the diameter is $(x-0)(x-0)+(y-\sqrt{2})(y+\sqrt{2})=0$
$\Rightarrow x^2+y^2=2$