Question

The locus of the centre of the circles which touch both the circles $x^2+y^2=a^2$ and $x^2+y^2=4ax$ externally is

• A. a circle
• B. a parabola
• C. an ellipse
• D. a hyperbola

Answer 1

Answer: (D)

Let, centre $=(h,k)$ and radius
$=r$ for the variable circle So,
using $C_1{C}_2=r_1+r_2$
for both cases we have:
$h^2+k^2=(r+a{)}^2\to (1)$ and
$(h-2a{)}^2+k^2=(r+2a{)}^2\to (2)$
Eq. (2) - Eq. (1), gives :
$r=\frac{a-4h}{2}\to$ (3)
Substitute (3) in (1) to get:
$12h^2-4k^2-24ah+9a^2=0$
$\therefore$ locus : $12x^2-4y^2-24ax+9a^2=0$ i.e. a hyperbola