Find the equation of the circle which passes through the points $\left(4,1\right)$ and $\left(6,5\right)$ and has its center on the line $4x + y - 16 = 0$.
Let the equation of the required circle be $\;$ $x^2 + y^2 + 2gx + 2fy + c = 0$ $\;\;\; \cdots \; (1)$
Equation $(1)$ passes through the point $\left(4,1\right)$.
$\therefore$ $\;$ We have $\;$ $\left(4\right)^2 + \left(1\right)^2 + 2 \times 4 \times g + 2 \times 1 \times f + c = 0$
i.e. $8g + 2f + 17 + c = 0$ $\;\;\; \cdots \; (2a)$
Equation $(1)$ also passes through the point $\left(6,5\right)$.
$\therefore$ $\;$ We have $\;$ $\left(6\right)^2 + \left(5\right)^2 + 2 \times 6 \times g + 2 \times 5 \times f + c = 0$
i.e. $12g + 10 f + 61 + c = 0$ $\;\;\; \cdots \; (2b)$
Solving equations $(2a)$ and $(2b)$ simultaneously we get,
$4g + 8f + 44 = 0$
i.e. $g + 2f + 11 = 0$ $\;\;\; \cdots \; (3a)$
Let the center of the required circle be $\left(-g, -f\right)$.
The center lies on $\;$ $4x + y - 16 = 0$
$\therefore$ $\;$ We have $\;$ $-4g - f - 16 = 0$ $\;\;\; \cdots \; (3b)$
Multiplying equation $(3b)$ with $2$ and adding to equation $(3a)$ gives
$-7g - 21 = 0$ $\implies$ $g = -3$
Substituting the value of $g$ in equation $(3a)$ gives
$-3 + 2f + 11 = 0$ $\implies$ $f = -4$
Substituting the values of $f$ and $g$ in equation $(2a)$ gives
$8 \times \left(-3\right) + 2 \times \left(-4\right) + 17 + c = 0$ $\implies$ $c = 15$
Substituting the values of $f$, $g$ and $c$ in equation $(1)$ gives the equation of the required circle as
$x^2 + y^2 - 6x - 8y + 15 = 0$