Question 124·Medium·Geometry and Trigonometry

Use the midpoint formula on $A(4,-2)$ and $B(10,6)$ to find the center $(h,k)$ of the circle.

Compute the distance between $A$ and $B$ to get the diameter. How do you get the radius from the diameter, and what does that tell you about $r^2$?

Once you know the center $(h,k)$ and $r^2$, plug them into $(x-h)^2 + (y-k)^2 = r^2$ and look for the answer choice with matching signs and constant term.

Enter the points A = (4,-2) and B = (10,6) in Desmos. You can type them as A=(4,-2) and B=(10,6) to see them on the graph.

In the expression list, type (4+10)/2 on one line and (-2+6)/2 on another. Desmos will show the simplified values; together they give you the coordinates of the circle’s center $(h,k)$.

Type sqrt((10-4)^2 + (6-(-2))^2) in Desmos. The result is the length of the diameter; divide this by 2 in another line to get the radius.

Using the values of $h$, $k$, and $r$ you found, mentally form $(x-h)^2 + (y-k)^2 = r^2$. Compare that structure—especially the signs on $h$ and $k$ and the value of $r^2$—with each answer choice to see which one matches.

In the coordinate plane, a circle with center $(h,k)$ and radius $r$ has equation

So we need to find the center and the radius of the circle whose diameter has endpoints $A(4,-2)$ and $B(10,6)$.

The endpoints of a diameter lie on opposite sides of the circle, so the center is the midpoint of segment $AB$.

Midpoint of $A(x_1,y_1)$ and $B(x_2,y_2)$ is

For $A(4,-2)$ and $B(10,6)$:

So $h=7$ and $k=2$ in the circle equation.

The length of $AB$ is the diameter. Use the distance formula between $A(4,-2)$ and $B(10,6)$:

This $10$ is the diameter, so the radius is half of that: $r = 10/2 = 5$, and thus $r^2 = 25$.

Substitute the center $(h,k) = (7,2)$ and $r^2 = 25$ into the standard form:

This matches answer choice D, so $(x-7)^2 + (y-2)^2 = 25$ is the correct equation of the circle.