Question

If the circles $x^2+y^2=c^2$ and $x^2+y^2+2ax=0$ touch each other, then

• A. they touch each other externally
• B. they touch each other internally
• C. $4c^2=a$
• D. $2a^2=c^2$

Answer 1

The correct option is B they touch each other internally

Given, $C_1:x^2+y^2=c^2$.
Here, the center is $C_1(0,0)$ and radius $r_1=c$.
Also, $C_2:x^2+y^2+2ax=0$
$\Rightarrow (x+a{)}^2+y^2=a^2$
Center $C_2(-a,0)$ and radius $r_2=a$
The distance between the centres of the two circles is given by $C_1{C}_2=a$.
Sum of the radii of the two circles $=a+c$.
As the sum of the two radii is greater than the distance between the centres of the two circles, the two circles cannot touch each other externally.
Hence, the two circles touch internally,
$\Rightarrow C_1{C}_2=c-a$
$2a=c$
$4a^2=c^2$
Hence, option B is correct