Question 732·Medium·Math

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The endpoints of a diameter of a circle in the $xy$ -plane are $(2,-3)$ and $(8,9)$ . Which equation represents this circle?

A circle described by the endpoints of a diameter has its center halfway between them and its radius equal to half their distance. Name the endpoints in Desmos, use midpoint for the center, then find the squared radius from distance. Put those values into the circle equation. Halving the squared distance instead of the distance makes the circle too large.

Hints

Hint 1
The midpoint is the point halfway between two endpoints. A diameter passes through the circle’s center, so its midpoint is the center. What does Desmos give when you enter the two endpoints and use midpoint?
Hint 2
The radius goes from the center to the circle, while the diameter goes all the way across. The endpoints’ distance is the diameter. How does halving that length change its square?
Hint 3
In a circle’s standard form, $(x-h)^2+(y-k)^2=r^2$ , the center is $(h,k)$ . The parentheses subtract the center’s coordinates, and the right side is the radius squared. Which values have you found for those parts?

Step-by-step

Find the center and squared radius with Desmos
Step 1
Locate the center
A diameter passes through the center, so its midpoint is the center. Type $A=(2,-3)$ and $B=(8,9)$ , then $\operatorname{midpoint}(A,B)$ . Click the plotted midpoint; Desmos shows $(5,3)$ . So the circle’s center is $(5,3)$ .
Step 2
Find the squared diameter
Call the diameter’s length $d$ and its square $q=d^2$ . Type $q=\operatorname{distance}(A,B)^2$ . Desmos shows $q=180$ : it squares and adds the horizontal change $8-2=6$ and the vertical change $9-(-3)=12$ . This is the squared diameter, not the squared radius.
Step 3
Turn the diameter into the squared radius
A radius is half a diameter, so $r^2=(d/2)^2=d^2/4=q/4$ . Halving a length divides its square by $4$ , not $2$ . Type $q/4$ ; Desmos shows $45$ for $r^2$ . Dividing $180$ by $2$ would give $90$ , but that would halve the square rather than the length.
Step 4
Write the circle equation
A circle with center $(h,k)$ has standard form $(x-h)^2+(y-k)^2=r^2$ . Use the center $(5,3)$ and $r^2=45$ :
$(x-5)^2+(y-3)^2=45.$
The minus signs put the center at $(5,3)$ . So this equation represents the circle. Choice C.

Question 732·Medium·Math

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