Two circles centred at \((2,3)\) and \((5,6)\) intersect each other. If the radii are equal, the equation of the common chord is

Let the radius of both circles are ' \(r\) '.
Now, equation of circle with centre at \((2,3)\) is
\[
S_{1} \equiv(x-2)^{2}+(y-3)^{2}=r^{2} \quad \text{...(i)}
\]
and equation of circle with centre at \((5,6)\) is
\[
S_{2} \equiv(x-5)^{2}+(y-6)^{2}=r^{2} \quad \text{...(ii)}
\]
Now, the equation common chord
\[
\begin{gathered}
\equiv \text { Radical axis of } \mathrm{S}_{1} \text { and } \mathrm{S}_{2}=0 \\
\equiv\left(\mathrm{S}_{1}-\mathrm{S}_{2}\right)=0 \\
\equiv\left[(\mathrm{x}-2)^{2}\right]+\left[(\mathrm{y}-3)^{2}\right] \\
\quad-[(\mathrm{x}-5)]^{2}-[(\mathrm{y}-6)]^{2}=0 \\
\equiv \mathrm{x}^{2}+\mathrm{y}^{2}+4-4 \mathrm{x}+9-6 \mathrm{x} \\
\quad-\mathrm{x}^{2}-\mathrm{y}^{2}-25-36+10 \mathrm{x}+12 \mathrm{y}=0 \\
\equiv 6 \mathrm{x}+6 \mathrm{y}-48=0 \\
\text { Common chord } \equiv \mathrm{x}+\mathrm{y}-8=0
\end{gathered}
\]