Question 176·Medium·Geometry and Trigonometry

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The endpoints of a diameter of a circle in the coordinate plane are $P(-2,3)$ and $Q(6,-1)$ . Which equation represents this circle?

When a circle question gives you the endpoints of a diameter, immediately think: (1) find the center using the midpoint formula, and (2) find $r^2$ by computing the distance squared between the endpoints and dividing by 4 (since the radius is half the diameter). Then plug the center and $r^2$ into $(x-h)^2 + (y-k)^2 = r^2$ and match the resulting form directly to the answer choices, which is often faster and less error-prone than computing the actual distance with a square root.

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Use the standard circle equation

Think about the standard form of a circle’s equation in the coordinate plane. What two key pieces of information do you need to plug into that formula?

Relate the diameter endpoints to the center

The endpoints $P(-2,3)$ and $Q(6,-1)$ are the ends of a diameter. How do you find the center of a segment when you know its endpoints?

Compute the midpoint carefully

Apply the midpoint formula to $P(-2,3)$ and $Q(6,-1)$ : average the $x$ -coordinates, and average the $y$ -coordinates. Watch out for signs when adding $-2$ and $6$ , and when adding $3$ and $-1$ .

Find the radius using the distance formula

Use the distance formula between $P$ and $Q$ to find the diameter length. Remember, the radius is half the diameter, and you only need $r^2$ , which can be found by taking one-fourth of the diameter squared.

Desmos Guide

Use Desmos to confirm the center (midpoint)

In Desmos, evaluate the midpoint coordinates separately: type (-2+6)/2 to get the $x$ -coordinate and (3+(-1))/2 to get the $y$ -coordinate. The pair of results is the center of the circle; compare this center to the $(h,k)$ values in each answer choice.

Use Desmos to compute the diameter squared and radius squared

To find the diameter squared, type (6 - (-2))^2 + (-1 - 3)^2 into Desmos and note the result. Then divide that result by 4 (for example, by typing previous_answer/4 or retyping the number /4) to get $r^2$ . Compare this $r^2$ value to the right side of each answer choice.

Match center and radius squared to a choice

Now look at the answer options and identify which equation uses the center you found in step 1 and the $r^2$ value you found in step 2. That option is the equation of the circle.

Step-by-step Explanation

Recall the standard form of a circle

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

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

So our job is to find the center $(h,k)$ and $r^2$ using the endpoints of the diameter.

Find the center as the midpoint of the diameter

The endpoints of the diameter are $P(-2,3)$ and $Q(6,-1)$ . The center of the circle is the midpoint of $PQ$ .

Use the midpoint formula: if the endpoints are $(x_1,y_1)$ and $(x_2,y_2)$ , then the midpoint is

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

Plug in the coordinates of $P$ and $Q$ :

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

So the center is $(2,1)$ , meaning $h=2$ and $k=1$ in the circle equation.

Find the radius squared using the distance between the endpoints

The distance between $P$ and $Q$ is the diameter of the circle. Let that distance be $d$ .

First find $d^2$ using the distance formula squared:

d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2

For $P(-2,3)$ and $Q(6,-1)$ :

(6 - (-2))^2 + (-1 - 3)^2 = 8^2 + (-4)^2 = 64 + 16 = 80

So $d^2 = 80$ .

The radius $r$ is half the diameter: $r = \dfrac{d}{2}$ . Therefore

r^2 = \left( \frac{d}{2} \right)^2 = \frac{d^2}{4} = \frac{80}{4} = 20

So $r^2 = 20$ .

Write the circle’s equation and match it to the choices

Now plug the center $(h,k) = (2,1)$ and $r^2 = 20$ into the standard form:

(x - 2)^2 + (y - 1)^2 = 20

This matches answer choice D, $(x-2)^2 + (y-1)^2 = 20$ .

Question 176·Medium·Geometry and Trigonometry

Strategy

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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