The endpoints of a diameter of a circle in the -plane are and . Which of the following is an equation of the circle?

Solution available with step-by-step explanation

SAT Circles Question 50 | Free SAT Prep | Aniko

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Question 50·Medium·Circles

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The endpoints of a diameter of a circle in the $xy$ -plane are $(-4,7)$ and $(6,-3)$ . Which of the following is an equation of the circle?

For circle questions with a given diameter, immediately find the center using the midpoint formula and then find the radius (or radius squared) from the distance between the endpoints. Use $r^2 = \dfrac{(x_2-x_1)^2 + (y_2-y_1)^2}{4}$ to get $r^2$ quickly in one step, then plug the center $(h,k)$ and $r^2$ into $(x-h)^2 + (y-k)^2 = r^2$ , and finally match your result to the given answer choices by checking both the center’s signs and the constant term.

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Use the endpoints of the diameter

What point lies exactly halfway between $(-4,7)$ and $(6,-3)$ ? That point is important for the circle.

Recall the midpoint and distance formulas

To find the point halfway between two points, use the midpoint formula. To find the length of the segment between them, use the distance formula.

Connect diameter and radius to the circle equation

Once you know the center and the radius, how do you plug those into the standard circle equation $(x-h)^2 + (y-k)^2 = r^2$ ?

Check both center and radius against the choices

Look at how each answer choice encodes the center and the radius squared. Which one has the correct center and the correct $r^2$ based on your calculations?

Desmos Guide

Plot the endpoints of the diameter

Enter the two points as separate lines: type (-4,7) on one line and (6,-3) on another. You should see the segment that will act as a diameter of the circle.

Check the midpoint (center) visually

In Desmos, use the midpoint feature by typing M = midpoint[(-4,7),(6,-3)]. The point M is the circle’s center. Compare its coordinates to the centers implied by the answer choices (from the expressions inside the parentheses).

Graph each answer choice’s circle

On separate lines, enter each option exactly as written, for example (x+1)^2 + (y+2)^2 = 50. Desmos will graph each circle. See which circle has its center at the midpoint M and contains both points $(-4,7)$ and $(6,-3)$ on its circumference.

Confirm the radius

Use the Desmos distance feature, e.g., type d = distance(M, (-4,7)) to find the radius from the center to an endpoint. Then compare $d^2$ to the $r^2$ value (the constant on the right side) in each answer choice to see which matches.

Step-by-step Explanation

Recall the standard form of a circle and relation to a diameter

A circle with center $(h,k)$ and radius $r$ has equation

(x-h)^2 + (y-k)^2 = r^2.

If you know the endpoints of a diameter, then:

The center of the circle is the midpoint of the diameter.
The diameter length is the distance between the endpoints, and the radius is half of that.

Find the center as the midpoint of the diameter

The endpoints of the diameter are $(-4,7)$ and $(6,-3)$ . Use the midpoint formula for points $(x_1,y_1)$ and $(x_2,y_2)$ :

\left(\frac{x_1+x_2}{2},\;\frac{y_1+y_2}{2}\right).

So the center is

\left(\frac{-4+6}{2},\;\frac{7+(-3)}{2}\right) = \left(\frac{2}{2},\;\frac{4}{2}\right) = (1,2).

Find the radius using the distance between the endpoints

First find the distance between $(-4,7)$ and $(6,-3)$ using the distance formula. For points $(x_1,y_1)$ and $(x_2,y_2)$ , the distance $d$ is

\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.

Here:

(6-(-4))^2 = 10^2 = 100,\\[0.7em] (-3-7)^2 = (-10)^2 = 100,

so the diameter length is

\sqrt{100+100} = \sqrt{200} = 10\sqrt{2}.

The radius is half the diameter:

r = \frac{10\sqrt{2}}{2} = 5\sqrt{2},

r^2 = (5\sqrt{2})^2 = 25\cdot 2 = 50.

Write the circle’s equation and match to a choice

We found the center is $(1,2)$ and $r^2 = 50$ . Plug these into the standard form $(x-h)^2 + (y-k)^2 = r^2$ with $h=1$ , $k=2$ :

(x-1)^2 + (y-2)^2 = 50.

This matches answer choice D, so the equation of the circle is $(x-1)^2 + (y-2)^2 = 50$ .

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation