Q.10. Find the equation of a circle which passes through the centre of the circle $x^2+y^2+8x+10y-7=0$ and is concentric with the circle $2x^2+2y^2-8x-12y-9=0$.

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} 2{\mathrm{x}}^2+2{\mathrm{y}}^2-8\mathrm{x}-12\mathrm{y}-9=0 \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2-4\mathrm{x}-6\mathrm{y}-\frac{9}{2}=0 \\ \mathrm{Centre} \mathrm{of} \mathrm{circle} \mathrm{is} \left(2,3\right) \\ \mathrm{As} \mathrm{circle} \mathrm{is} \mathrm{concentric} \mathrm{with} {\mathrm{x}}^2+{\mathrm{y}}^2-4\mathrm{x}-6\mathrm{y}-\frac{9}{2}=0, \mathrm{hence} \\ \mathrm{centre} \mathrm{of} \mathrm{required} \mathrm{circle} \mathrm{is} \mathrm{also} \mathrm{O}\left(2,3\right) \\ \mathrm{Centre} \mathrm{of} \mathrm{circle} {\mathrm{x}}^2+{\mathrm{y}}^2+8\mathrm{x}+10\mathrm{y}-7=0 \mathrm{is} \mathrm{A}\left(-4,-5\right) \\ \mathrm{Required} \mathrm{circle} \mathrm{passes} \mathrm{through} \mathrm{A}\left(-4,-5\right) \\ \mathrm{Hence} \mathrm{radius}=\mathrm{OA}=\sqrt{{\left(2+4\right)}^2+{\left(3+5\right)}^2}=\sqrt{36+64}=10 \\ \mathrm{Hence} \mathrm{equation} \mathrm{is} \\ {\left(\mathrm{x}-2\right)}^2+{\left(\mathrm{y}-3\right)}^2={10}^2 \\ \Rightarrow {\left(\mathrm{x}-2\right)}^2+{\left(\mathrm{y}-3\right)}^2=100 \end{gathered}$$

Hope this information will clear your doubts about this topic.

If you have any doubts just ask here on the ask and answer forum and our experts will try to help you out as soon as possible.
Regards