A circle in the $xy$-plane has center $(5,3)$, and the point $(10,-9)$ is on the circle. Which of the following is an equation of the circle?

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We need to find the standard form of the circle.

$$(x-h)^2+(y-k)^2=r^2, \text{ where } (h,k) \text{ is the center and } r \text{ is the radius}$$ We are given the center $(5,3)$ but must find the square of the radius through distance formula or pythagorean theorem. $$r^2=(x_1-x_2)^2+(y_1-y_2)^2$$ $$r^2= (x_1-x_2)^2+(y_1-y_2)^2$$ $$r^2 = (5-10)^2+(3-(-9))^2$$ $$r^2 = 25+144$$ $$r^2=169$$

Therefore,

$$(x-\colorbox{aqua}{$5$})^2+(y-\colorbox{aqua}{$3$})^2=\colorbox{aqua}{$169$}$$ $$= \boxed{(x-5)^2+(y-3)^2=169 }$$