Find the equation of the circle passing through the points (2,–3), (–4,5) and having the centre on the line 4x + 3y + 1 = 0

Given points are A(2, –3) and B(–4, 5) centre of the required circle lies on the line 4x + 3y +1=  0 . Required circle passes through A and B, clearly A, B satisfy equation the given line
$\Rightarrow$A, B lie on the given line and also lie on the circle

$\Rightarrow$A, B are ends of a diameter of the circle
$\therefore$Equation of the required circle is
$\begin{array}{l}\left(\mathrm{x}-{\mathrm{x}}_1\right)\left(\mathrm{x}-{\mathrm{x}}_2\right)+\left(\mathrm{y}-{\mathrm{y}}_1\right)\left(\mathrm{y}-{\mathrm{y}}_2\right)=0\\ \Rightarrow (\mathrm{x}-2)(\mathrm{x}+4)+(\mathrm{y}+3)(\mathrm{y}-5)=0\\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{x}-2\mathrm{y}-23=0\end{array}$