AIEEE 2011 (Offline)Question 1
The two circles $x^{2}+y^{2}=ax$ and $x^{2}+y^{2}=c^{2}$ $(c>0)$ touch each other if :
Circle
Solution
Concept. The first circle passes through the origin, the centre of the second, so it can only touch the second internally. Formulas used. (1) Internal contact: $d=|r_1-r_2|$.Step 1 - Circles. $x^2+y^2-ax=0$: centre $\left(\frac a2,0\right)$, $r_1=\frac{|a|}2$. $x^2+y^2=c^2$: centre $(0,0)$, $r_2=c$. Step 2 - Distance. $d=\frac{|a|}2$. External contact would need $\frac{|a|}2=\frac{|a|}2+c$, impossible as $c>0$. Step 3 - Use (1). $$\frac{|a|}2=c-\frac{|a|}2\Rightarrow|a|=c.$$ Answer: A