Question

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

• A. circle
• B. parabola
• C. hyperbola
• D. none of these

Answer 1

Answer: (C)

hyperbola

Let $C_1$ and $C_2$ be the centres of two circles of
radii $r_1$ and $r_2$ respectively. Let $P$ be
the centre and $r$ be the radii of a circle touching the two given
circles externally, then
$C_{1}P=r+r_1$ and $C_{2}P=r+r_2$
$\Rightarrow C_{1}P-C_{2}P=r_1-r_2=$ constant.
point $P$ moves in such a way that the difference of its distances
from two fixed points $C_1$ and $C_2$ is constant.
Therefore, $P$ describes a hyperbola whose two foci are at $C_1$
and $C_2$
Hence, option 'C' is correct.