Question

The locus of the centres of the circles which touch the given two circles externally is

• A. Pair of lines
• B. Ellipse
• C. Hyperbola
• D. Circle

Answer 1

Answer: (C)

Hyperbola

$\therefore r_1+r=\sqrt{(h+g_1{)}^2+(k+f_1{)}^2}--\left(1\right)$
$r_2+r=\sqrt{(h+g_2{)}^2+(k+f_2{)}^2}--\left(2\right)$
from (2) & (1)
$\Rightarrow (r_2-r_1)=\sqrt{(h+g_2{)}^2+(k+f_2{)}^2}-\sqrt{(h+g_1{)}^2+(k+f_1{)}^2}$
So, difference of distances from two points $(-g_1-f_1)$ and
$(-g_2-f_2)$ is constant $(r_2-r_1)$ is a locus of
hyperbola.