Question 40·Medium·Circles

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A circle in the coordinate plane has endpoints of a diameter at $(2,-1)$ and $(-6,5)$ . What is an equation of this circle?

For circle problems with endpoints of a diameter, immediately write down the standard form $(x-h)^2+(y-k)^2=r^2$ , then compute the center as the midpoint of the two endpoints and the diameter length using the distance formula. Halve the diameter to get the radius and square it for $r^2$ , plug $h$ , $k$ , and $r^2$ into the formula, and finally match your equation carefully to the choices, paying close attention to the signs that determine the center.

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Identify what defines a circle

To write an equation of a circle in the coordinate plane, what two pieces of information do you need, and what is the standard form of the equation that uses them?

Use the endpoints of the diameter

The two given points are endpoints of a diameter. How can you use these two points to find the center of the circle?

Relate diameter to radius

Once you know the distance between the two given points, what does that distance represent, and how do you turn it into the radius (and then $r^2$ ) for the circle equation?

Desmos Guide

Plot the diameter endpoints

Enter the two points as (2,-1) and (-6,5) in Desmos so you can see the segment that is supposed to be a diameter of the circle.

Graph each answer choice as a circle

On separate lines, type each option exactly as written, for example y^2+4y+x^2-4x=10 after expanding, or more simply use implicit form like (x-2)^2+(y+2)^2=10, so that each choice appears as a circle on the graph.

Check which circle fits the given diameter

Look for the circle on which both points (2,-1) and (-6,5) lie, and such that the line segment between these points passes through the circle’s center (looks like a straight line cutting the circle exactly in half). The equation of that circle matches the correct answer choice.

Step-by-step Explanation

Recall the standard circle equation

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

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

So we need to find the center $(h,k)$ and the radius $r$ from the endpoints of the diameter $(2,-1)$ and $(-6,5)$ .

Find the center as the midpoint of the diameter

The center of a circle is the midpoint of any diameter. 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).

Here, $(x_1,y_1) = (2,-1)$ and $(x_2,y_2) = (-6,5)$ , so

\text{center } (h,k) = \left(\frac{2+(-6)}{2},\;\frac{-1+5}{2}\right) = \left(\frac{-4}{2},\;\frac{4}{2}\right) = (-2,2).

So $h=-2$ and $k=2$ .

Find the radius from the diameter

The distance between the two endpoints of the diameter is the length of the diameter. Use the distance formula:

\text{diameter length } d = \sqrt{(2 - (-6))^2 + (-1 - 5)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10.

The radius is half the diameter, so

r = \frac{10}{2} = 5, \quad r^2 = 25.

Write the equation of the circle and match the choice

We have center $(-2,2)$ and $r^2 = 25$ . Substitute into $(x-h)^2 + (y-k)^2 = r^2$ :

(x - (-2))^2 + (y - 2)^2 = 25,

which simplifies to

(x+2)^2 + (y-2)^2 = 25.

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

Question 40·Medium·Circles

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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