Question 116·Medium·Geometry and Trigonometry

Compute $(\Delta x)^2+(\Delta y)^2$ from $C$ to $A$ to get $r^2$ (you do not need to take a square root).

Plug the center $(h,k)$ and $r^2$ into $(x-h)^2+(y-k)^2=r^2$.

Enter the points $C=(2,-1)$ and $A=(-4,3)$ so you can refer to them on the graph.

Enter each answer choice as an equation (one at a time or all together). Look for the circle whose center matches $C$ and that passes through $A$.

For the circle that fits, use the distance from $C$ to $A$ (visually or by calculation) to check that its radius squared matches the constant on the right side.

From the graph, the center is $C(2,-1)$ and the circle passes through $A(-4,3)$. So the radius is the distance from $C$ to $A$.

Compute the horizontal and vertical changes from $C$ to $A$:

$\Delta x=-4-2=-6$ and $\Delta y=3-(-1)=4$.

So

A circle with center $(h,k)$ and radius $r$ has equation $(x-h)^2+(y-k)^2=r^2$.

Here $(h,k)=(2,-1)$ and $r^2=52$, so the equation is $(x-2)^2+(y+1)^2=52$.