Question 700·Medium·Math

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In the $xy$ -plane, the line segment with endpoints $A(-6,5)$ and $B(2,-3)$ is a diameter of a circle. Which equation represents this circle?

The phrase endpoints of a diameter tells you how to build the circle: a diameter runs through the center, so its midpoint is the center. Name the endpoints in Desmos and use midpoint to find it. Then square half the distance between the endpoints to get the number on the equation’s right. Watch the signs inside the parentheses.

Hints

Hint 1
A midpoint is the point halfway between two endpoints. Because a diameter passes through the center, its midpoint is the center. What will midpoint(A,B) give you after you enter the two points?
Hint 2
The distance from one endpoint to the other is the whole diameter. A radius reaches only from the center to the circle, so halve that distance before you square it for the equation.
Hint 3
A circle’s standard form is $(x-h)^2+(y-k)^2=r^2$ , where $(h,k)$ is its center. The parentheses subtract the center’s coordinates. What happens inside the $x$ -parenthesis when $h$ is negative?

Step-by-step

Find the center and squared radius
Step 1
Find the center
Enter $A=(-6,5)$ and $B=(2,-3)$ in Desmos, then type $\operatorname{midpoint}(A,B)$ . Click the plotted midpoint, the point halfway between the endpoints. Desmos shows $(-2,1)$ . That’s the center because the given segment is a diameter.
Step 2
Find the squared radius
The distance between $A$ and $B$ is the diameter, which is twice the radius. Halve the distance before squaring it: $r^2=\left(\frac{AB}{2}\right)^2$ . Add $\left(\operatorname{distance}(A,B)/2\right)^2$ in Desmos. It prints $32$ , so $r^2=32$ .
Step 3
Put the values into circle form
A circle’s standard form is $(x-h)^2+(y-k)^2=r^2$ , where $(h,k)$ is the center. Substitute the center $(-2,1)$ and $r^2=32$ :
$\begin{aligned} (x-h)^2+(y-k)^2&=r^2\\[1.4em] (x-(-2))^2+(y-1)^2&=32. \end{aligned}$
Step 4
Simplify the center’s negative coordinate
Simplify $x-(-2)$ to $x+2$ ; the plus sign still represents a center with $x$ -coordinate $-2$ :
$(x+2)^2+(y-1)^2=32.$
That equation represents the circle with the given diameter. Choice D.

Question 700·Medium·Math

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