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10-1.Circle and System of Circles
hard
Two circle ${x^2} + {y^2} = ax$ and ${x^2} + {y^2} = {c^2}$ touch each other if
A
$|a|=c$
B
$a=2c$
C
$|a|=2c$
D
$2|a|=c$
(AIEEE-2011)
Solution
The centres and radii are
$\left(x-\frac{a}{2}\right)^{2}+y^{2}=\frac{a^{2}}{4}, \quad x^{2}+y^{2}=c^{2}$
Centre $\left(\frac{a}{2}, 0\right)$ and $(0,0)$ and radius $=\frac{a}{2}$ and $c$
$\sqrt{\left(\frac{a}{2}\right)^{2}+(0-0)}=|| \frac{a}{2}|\pm c|$
$ \Rightarrow\left|\frac{a}{2}\right|=|| \frac{a}{2}|\pm c|$
$\Rightarrow\left|\frac{a}{2}\right|=c-\left|\frac{a}{2}\right|, \quad \therefore|a|=c$
Standard 11
Mathematics
