Question
The two circles $${x^2} + {y^2} = ax$$ and $${x^2} + {y^2} = {c^2}\left( {c > 0} \right)$$ touch each other if-
A.
$$\left| a \right| = c$$
B.
$$a = 2c$$
C.
$$\left| a \right| = 2c$$
D.
$$2\left| a \right| = c$$
Answer :
$$\left| a \right| = c$$
Solution :
As centre of one circle is (0, 0) and other circle passes through (0, 0), therefore
$$\eqalign{
& {\text{Also }}{C_1}\left( {\frac{a}{2},\,0} \right)\,\,\,\,{C_2}\left( {0,\,0} \right) \cr
& {r_1} = \frac{a}{2}\,\,\,\,{r_2} = C \cr
& {C_1}{C_2} = {r_1} - {r_2} = \frac{a}{2} \cr
& \Rightarrow C - \frac{a}{2} = \frac{a}{2} \cr
& \Rightarrow C = a \cr} $$
If the two circles touch each other, then they must touch each other internally.