Find the equation to the circle which passes through the points $\left(-1, 0\right)$, $\left(3, 4\right)$ and has its center on the line $\;$ $3x - y - 23 = 0$
Let the equation of the required circle be $\;$ $x^2 + y^2 + 2gx + 2fy + c = 0$ $\;\;\; \cdots \; (1)$
$\therefore \;$ Center of the required circle $= \left(-g, -f\right)$
Given: Center lies on the line $\;$ $3x - y - 23 = 0$
$\therefore \;$ We have $\;$ $- 3g + f - 23 = 0$ $\;\;\; \cdots \; (2)$
The required circle passes through the points $\;$ $\left(-1, 0\right)$ $\;$ and $\;$ $\left(3, 4\right)$
$\therefore \;$ We have from equation $(1)$,
$\left(-1\right)^2 = 0^2 + 2 \times g \times \left(-1\right) + 2 \times f \times 0 + c = 0$
i.e. $\;$ $- 2 g + c + 1 = 0$
$\implies$ $g = \dfrac{c + 1}{2}$ $\;\;\; \cdots \; (3)$
and $\;$ $3^2 + 4^2 + 2 \times g \times 3 + 2 \times f \times 4 + c = 0$
i.e. $\;$ $9 + 16 + 6g + 8f + c = 0$
i.e. $\;$ $6g + 8f + c + 25 = 0$ $\;\;\; \cdots \; (4)$
Substituting the value of $g$ from equation $(3)$ in equation $(4)$ gives
$6 \left(\dfrac{c + 1}{2}\right) + 8 f + c + 25 = 0$
i.e. $\;$ $3c + 3 + 8f + c + 25 = 0$
i.e. $\;$ $8f + 4c + 25 = 0$
$\implies$ $f = \dfrac{-c - 7}{2}$ $\;\;\; \cdots \; (5)$
In view of equations $(4)$ and $(5)$, equation $(2)$ becomes,
$-3 \left(\dfrac{c + 1}{2}\right) - \left(\dfrac{c + 7}{2}\right) - 23 = 0$
i.e. $\;$ $\dfrac{-3c}{2} - \dfrac{3}{2} - \dfrac{c}{2} - 23 = 0$
i.e. $\;$ $-2c - 5 - 23 = 0$
i.e. $\;$ $2c = -28$ $\implies$ $c = -14$
Substituting the value of $c$ in equations $(3)$ and $(5)$ gives
$g = \dfrac{-14 + 1}{2} = \dfrac{-13}{2}$ $\;$ and $\;$ $f = \dfrac{14 - 7}{2} = \dfrac{7}{2}$
$\therefore \;$ The equation of the required circle is
$x^2 + y^2 + 2 \times \left(\dfrac{-13}{2}\right) \times x +2 \times \dfrac{7}{2} \times y - 14 = 0$
i.e. $\;$ $x^2 + y^2 - 13x + 7y -14 = 0$