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To find the standard form of the equation of a circle given the endpoints of the diameter, we can use the following approach:

1. **Determine the center**: The center of the circle is the midpoint of the diameter. The midpoint formula is:

\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Given points \( (1, 2) \) and \( (-3, 5) \), the center \( (h, k) \) is:

\[
h = \frac{1 + (-3)}{2} = \frac{-2}{2} = -1
\]
\[
k = \frac{2 + 5}{2} = \frac{7}{2} = 3.5
\]

So, the center of the circle is \( (-1, 3.5) \).

2. **Find the radius**: The radius is half the length of the diameter. To find the length of the diameter, use the distance formula between the two endpoints:

\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substitute the values of \( (x_1, y_1) = (1, 2) \) and \( (x_2, y_2) = (-3, 5) \):

\[
\text{Distance} = \sqrt{((-3 - 1)^2 + (5 - 2)^2)} = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]

Since the distance represents the diameter, the radius \( r \) is half of the diameter:

\[
r = \frac{5}{2} = 2.5
\]

3. **Write the standard form**: The standard form of the equation of a circle is:

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

Substitute the center \( (h, k) = (-1, 3.5) \) and radius \( r = 2.5 \):

\[
(x + 1)^2 + (y - 3.5)^2 = (2.5)^2
\]
\[
(x + 1)^2 + (y - 3.5)^2 = 6.25
\]

Thus, the standard form of the equation of the circle is:

\[
(x + 1)^2 + (y - 3.5)^2 = 6.25
\]

Would you like any more details or explanations?

### Here are 5 related questions:
1. How do you find the equation of a circle if only the center and radius are given?
2. What is the general form of the equation of a circle?
3. How can you determine the equation of a circle given three points on the circle?
4. What is the relationship between the distance formula and the radius of a circle?
5. How would the equation change if the radius were doubled?

### Tip:
The equation of a circle can be converted between standard form and general form by expanding and simplifying terms. Keep practicing both forms!

Write the standard form of the equation of the circle with endpoints of a diameter at the points (1, 2) and (-3, 5).

Learn how to find the standard form of the equation of a circle with diameter endpoints at (1, 2) and (-3, 5). This problem involves applying the midpoint formula to find the center and the distance formula to calculate the radius. Suitable for students in Grades 9-10.

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