Question

Locus of the centre of a variable circle which touches the circles $x^2+y^2-2x-4y-20=0$ internally and $x^2+y^2+4x-2y+4=0$ externally, is

• A. a circle whose radius is $4$ units
• B. a circle whose radius is $6$ units
• C. an ellipse whose major axis length is $4$ units
• D. an ellipse whose major axis length is $6$ units

Answer 1

The correct option is D an ellipse whose major axis length is $6$ units

$S_1:x^2+y^2-2x-4y-20=0$
$\Rightarrow C_1\equiv (1,2),r_1=5$
$S_2:x^2+y^2+4x-2y+4=0$
$\Rightarrow C_2\equiv (-2,1),r_2=1$

$C_1{C}_2=\sqrt{10}<|r_1-r_2|$
So, $S_2=0$ lies inside the circle $S_1=0$


Let $C(h,k)$ be centre and $r$ be the radius of the variable circle.
From the figure,
$C{C}_1=r_1-r\dots \left(1\right)$
$C{C}_2=r_2+r\dots \left(2\right)$
$\left(1\right)+\left(2\right),$ we get
$C{C}_1+C{C}_2=r_1+r_2=6$
So, locus of $C$ is an ellipse whose foci are $C_1$ and $C_2.$
Length of major axis $=2a=6$ units.