The circle through the two points (–2, 5), (0, 0) and intersecting x² + y² – 4x + 3y – 1 = 0 orthogonally is

circle s = 0 passes through (0, 0) and $(-2,5)\Rightarrow \mathrm{C}=0\text{ and }-4\mathrm{g}+10\mathrm{f}=-29$
S = 0 cuts given circle orthogonally
$\begin{array}{l}\Rightarrow -4\mathrm{g}+3\mathrm{f}=-1\text{ solving }\mathrm{f}=-4,\mathrm{g}=\frac{-11}{4}\\ \therefore \text{ circle is }2{\mathrm{x}}^2+2{\mathrm{y}}^2-11\mathrm{x}-16\mathrm{y}=0\end{array}$