Writing equations of circles Worksheets - Free Printable

We are given the endpoints of a diameter of a circle:
$$
(13, -10) \quad \text{and} \quad (-1, 14)
$$

We are to find the equation of the circle, and choose the correct option among A, B, C, D.

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Step 1: Find the center of the circle

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

Use the midpoint formula:
$$
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
$$

Plug in the points:
$$
\left( \frac{13 + (-1)}{2}, \frac{-10 + 14}{2} \right) = \left( \frac{12}{2}, \frac{4}{2} \right) = (6, 2)
$$

So, the center is at $ (6, 2) $

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Step 2: Find the radius

The radius is half the length of the diameter. First, compute the distance between the two endpoints:

Use the distance formula:
$$
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

$$
= \sqrt{(-1 - 13)^2 + (14 - (-10))^2} = \sqrt{(-14)^2 + (24)^2} = \sqrt{196 + 576} = \sqrt{772}
$$

So, the diameter is $ \sqrt{772} $, and the radius is $ \frac{\sqrt{772}}{2} $

But we don't need the radius directly — we need the square of the radius for the equation.

$$
r^2 = \left( \frac{\sqrt{772}}{2} \right)^2 = \frac{772}{4} = 193
$$

✔ So, $ r^2 = 193 $

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Step 3: Write the equation of the circle

General form:
$$
(x - h)^2 + (y - k)^2 = r^2
$$
where $ (h, k) $ is the center.

We have:
- $ h = 6 $
- $ k = 2 $
- $ r^2 = 193 $

So:
$$
(x - 6)^2 + (y - 2)^2 = 193
$$

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Step 4: Match with options

Let’s check each choice:

A) $ (x + 3)^2 + (y + 5)^2 = 193 $ → Center: $ (-3, -5) $ ✘
B) $ x^2 + (y + 4)^2 = 37249 $ → Center: $ (0, -4) $, $ r^2 = 37249 $ ✘
C) $ (x - 6)^2 + (y - 2)^2 = 193 $ → ✔ Matches our result!
D) $ (x + 2)^2 + (y - 6)^2 = 193 $ → Center: $ (-2, 6) $ ✘

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✔ Final Answer:

C) $ (x - 6)^2 + (y - 2)^2 = 193 $

This is the correct equation of the circle with diameter endpoints $ (13, -10) $ and $ (-1, 14) $.