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Q.
Equation of the circle which passes through the intersection of $x^{2}+y^{2}+13 x-3 y=0$ and $2 x^{2}+2 y^{2}+4 x-7 y-25=0$
whose centre lies on $13 x+30 y=0$ is
Conic Sections
Solution:
The equation of required circle is $s _{1}+\lambda s _{2}=0$
$=x^{2}(1+\lambda)+y^{2}(1+\lambda)+x(2+13 \lambda)-y$
$\left(\frac{7}{2}+3 \lambda\right)-\frac{25}{2}=0$
Centre $=\left(\frac{-(2+13 \lambda)}{2}, \frac{7 / 2+3 \lambda}{2}\right)$
$\because$ Centre lies on $13 x+30 y=0$
$\Rightarrow -13\left(\frac{2+13 \lambda}{2}\right)+30\left(\frac{7 / 2+3 \lambda}{2}\right)=0 $ $\Rightarrow \lambda=1$