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(A) Step-$1$: Find the radius $r$ of the circle.The radius is the distance between the center $(2, -1)$ and the point $(3, 6)$ on the circle.Using the

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$x^2 + y^2 - 4x + 2y - 45 = 0$

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(A) Step-$1$: Find the radius $r$ of the circle.The radius is the distance between the center $(2, -1)$ and the point $(3, 6)$ on the circle.Using the distance formula $r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$:$r = \sqrt{(3 - 2)^2 + (6 - (-1))^2}$$r = \sqrt{(1)^2 + (7)^2} = \sqrt{1 + 49} = \sqrt{50}$.Step-$2$: Use the standard equation of a circle $(x - h)^2 + (y - k)^2 = r^2$,where $(h, k)$ is the center.$(x - 2)^2 + (y - (-1))^2 = (\sqrt{50})^2$$(x - 2)^2 + (y + 1)^2 = 50$$x^2 - 4x + 4 + y^2 + 2y + 1 = 50$$x^2 + y^2 - 4x + 2y + 5 = 50$$x^2 + y^2 - 4x + 2y - 45 = 0$.

Find the equation of the circle passing through the point $(3, 6)$ and having its center at $(2, -1)$.

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