If the coordinates of the endpoints of a diameter of the circle are​ known, the equation of a circle can be found.​ First, find the midpoint of the​ diameter, which is the center of the circle. Then find the​ radius, which is the distance from the center to either endpoint of the diameter. Finally use the​ center-radius form to find the equation.

Find the​ center-radius form for the circle having the endpoints 

left parenthesis 3 comma 6 right parenthesis(3,6) 

and 

left parenthesis negative 7 comma 2 right parenthesis(−7,2) 

of a diameter.




To find the equation of the circle based on the endpoints of its diameter, follow these steps:

### 1. Find the midpoint of the diameter (center of the circle):
The midpoint formula is:
\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Given the endpoints \((3, 6)\) and \((-7, 2)\):
\[
\text{Midpoint} = \left( \frac{3 + (-7)}{2}, \frac{6 + 2}{2} \right) = \left( \frac{-4}{2}, \frac{8}{2} \right) = (-2, 4)
\]
Thus, the center of the circle is \((-2, 4)\).

### 2. Find the radius:
The radius is the distance from the center \((-2, 4)\) to either endpoint of the diameter. Using the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
For the distance from the center \((-2, 4)\) to the endpoint \((3, 6)\):
\[
\text{Distance} = \sqrt{(3 - (-2))^2 + (6 - 4)^2} = \sqrt{(3 + 2)^2 + (6 - 4)^2} = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29}
\]
So the radius is \(\sqrt{29}\).

### 3. Write the equation of the circle in center-radius form:
The standard form of a circle's equation is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Where \((h, k)\) is the center and \(r\) is the radius. Substituting the center \((-2, 4)\) and the radius \(\sqrt{29}\):
\[
(x + 2)^2 + (y - 4)^2 = (\sqrt{29})^2
\]
\[
(x + 2)^2 + (y - 4)^2 = 29
\]

Thus, the equation of the circle is:
\[
(x + 2)^2 + (y - 4)^2 = 29
\]

### Would you like further details or clarifications?

Here are five related questions for further exploration:
1. How do you find the equation of a circle if the radius and center are given directly?
2. Can you determine the equation of a circle given three non-collinear points?
3. How do you find the length of the diameter from the equation of a circle?
4. What is the general form of the equation of a circle?
5. How do you convert the general form of a circle equation to center-radius form?

**Tip:** Always check that the radius you calculate is consistent by verifying with both endpoints of the diameter!

Find the center-radius form for the circle having the endpoints (3,6) and (-7,2) of a diameter.

Learn how to find the equation of a circle given the endpoints of its diameter at (3,6) and (-7,2). This guide covers how to calculate the center using the midpoint formula and the radius using the distance formula, suitable for Grades 9-11.

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