Question 716·Medium·Math

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In the coordinate plane shown, circle $k$ has center $C$ and passes through point $A$ .

Which choice is an equation of circle $k$ ?

A circle shown with its center and a point on its edge can be written in center-radius form: $(x-h)^2+(y-k)^2=r^2$ , where $(h,k)$ is the center. Read the center from the graph, then square and add the horizontal and vertical changes to the edge point to find $r^2$ . Watch the signs inside the parentheses: they are opposite the center coordinates.

Hints

Hint 1
In center-radius form, each parenthesis subtracts a coordinate of the center. Read $C$ from the grid. What should you subtract from $x$ and $y$ so both parentheses become zero at the center?
Hint 2
A radius runs from the center to a point on the circle. Find the horizontal and vertical changes from $C$ to $A$ by subtracting the $x$ -coordinates and the $y$ -coordinates separately.
Hint 3
The horizontal and vertical changes make perpendicular sides of a right triangle. The Pythagorean theorem says their squares add to the squared radius. What number do you get when you square and add those changes?

Step-by-step

Build the equation from the center and radius
Step 1
Put the center into circle form
The graph places $C$ at $(2,-1)$ . In center-radius form, $(x-h)^2+(y-k)^2=r^2$ , the center is $(h,k)$ . Substitute the center coordinates:
$(x-2)^2+(y-(-1))^2=r^2.$
Simplify the double negative:
$(x-2)^2+(y+1)^2=r^2.$
The plus sign in $y+1$ comes from subtracting the center's $y$ -coordinate, $-1$ .
Step 2
Find the changes from the center to the circle
The graph places $A$ at $(-4,3)$ . Since $A$ is on the circle, $CA$ is a radius, not a diameter. Subtract matching coordinates to find the horizontal and vertical changes:
$-4-2=-6,\qquad 3-(-1)=4.$
Step 3
Find the squared radius
The horizontal and vertical changes are perpendicular, so their squares add to $r^2$ . Type $(-6)^2+4^2$ in Desmos. It shows $52$ , so the squared radius is $r^2=52$ . You don't need to take a square root: the circle equation uses $r^2$ .
Step 4
Complete the circle equation
Put $r^2=52$ into the form you built for center $C$ . The equation of circle $k$ is
$(x-2)^2+(y+1)^2=52.$
Choice A.

Question 716·Medium·Math

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