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Q.
For what value of $k$, does the equation$9 x^{2}+y^{2}=k\left(x^{2}-y^{2}-2 x\right)$represent equation of a circle?
Conic Sections
Solution:
The given equation $9 x^{2}+y^{2}=k\left(x^{2}-y^{2}-2 x\right)$ can be written as
$9 x^{2}+y^{2}-k x^{2}+k y^{2}+2 k x=0$
$\Rightarrow (9-k) x^{2}+(1+k) y^{2}+2 k x=0$
This equation represents a circle, if coefficients of $x^{2}$ and $y^{2}$ are equal.
so, $9-k=1+k $
$\Rightarrow 2 k=8 $
$ \Rightarrow k=4$