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Q.
The two circles $x^{2}+y^{2}=ax$ and $x^{2}+y^{2}=c^{2} \, \left(c > 0\right)$ touch each other, if $\left|\frac{c}{a}\right|$ is equal to
NTA AbhyasNTA Abhyas 2020Conic Sections
Solution:
The two circles touch each other internally as shown in the figure below
For the circle $x^{2}+y^{2}-2ax=0$ , the centre is at $\left(\frac{a}{2} , 0\right)$ and radius is $\frac{a}{2}$ units.
Also, the circle is passing through the origin.
For the circle $x^{2}+y^{2}=a^{2}$ , the centre is at $\left(0 , 0\right)$ and radius is $c$ units
From the figure,
$c=2\left|\frac{a}{2}\right|=\left|a\right|$
Hence, $\left|\frac{c}{a}\right|=1$
