Question

The locus of the center of a circle which touches two given circles externally is

• A. an ellipse
• B. a parabola
• C. a hyperbola
• D. none of the above

Answer 1

Answer: (C)

a hyperbola

Let $S$ and $S^{\prime }$ be the centres $a$ and $b$ be the radii of the given circles.

Let $P$ be the centre and $r$ be the radius of the circle which touches the given circles externally.

Then, $S^{\prime }P=r+a$ and $SP=r+b$

$\therefore S^{\prime }P=-SP=(r+a)-(r+b)=a-b=$ constant.

Hence, the locus of $P$ is a hyperbola whose foci are $S$ and $S^{\prime }.$