Question

A circle which made an intercept of $2a$ on $y-$axis and $x-$axis is the tangent to it. The locus of the center of the circle is

• A. Hyperbola with eccentricity $\sqrt{2}$
• B. Hyperbola with eccentricity $\sqrt{3}$
• C. Ellipse with eccentricity $\frac{1}{\sqrt{2}}$
• D. Parabola with focus $(0,2a)$

Answer 1

The correct option is A Hyperbola with eccentricity $\sqrt{2}$

Let the center of the circle is $(h,k)$
$\therefore$ equation of the circle is
$(x-h{)}^2+(y-k{)}^2=k^2{x}^2+y^2+h^2-2hx-2ky=0$
For $y$ intercept of the circle
$y^2+h^2-2ky=0$
$y=k\pm \sqrt{k^2-h^2}$
length of the $y$ intercept
$2\sqrt{k^2-h^2}=2a$
$\Rightarrow k^2-h^2=a^2$
Locus will be $y^2-x^2=a^2$