Find the equation of a circle in standard form where C(2, -1) and D(-1, -3) are 
 endpoints of a diameter

To find the equation of a circle in standard form, we use the formula:

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

where \((h, k)\) is the center of the circle and \(r\) is the radius.

### Step 1: Find the center
The center of a circle is the midpoint of its diameter. Given that \(C(2, -1)\) and \(D(-1, -3)\) are endpoints of the diameter, we find the midpoint of these two points:

\[
h = \frac{x_1 + x_2}{2} = \frac{2 + (-1)}{2} = \frac{1}{2}
\]
\[
k = \frac{y_1 + y_2}{2} = \frac{-1 + (-3)}{2} = \frac{-4}{2} = -2
\]

So, the center \((h, k)\) is \(\left( \frac{1}{2}, -2 \right)\).

### Step 2: Find the radius
The radius is half the length of the diameter. The length of the diameter is the distance between \(C(2, -1)\) and \(D(-1, -3)\), which is calculated using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
\[
d = \sqrt{(-1 - 2)^2 + (-3 - (-1))^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]

The radius \(r\) is half of this distance:
\[
r = \frac{\sqrt{13}}{2}
\]

### Step 3: Write the equation
Now that we have the center \(\left( \frac{1}{2}, -2 \right)\) and the radius \(r = \frac{\sqrt{13}}{2}\), we can write the equation of the circle in standard form:

\[
\left(x - \frac{1}{2}\right)^2 + (y + 2)^2 = \left(\frac{\sqrt{13}}{2}\right)^2
\]
\[
\left(x - \frac{1}{2}\right)^2 + (y + 2)^2 = \frac{13}{4}
\]

This is the standard form of the equation of the circle.

Would you like further details or have any questions?

### Related Questions:
1. How do you find the equation of a circle if only the radius and center are given?
2. What is the process for converting the general form of a circle’s equation to standard form?
3. How can we find the center of a circle if the endpoints of its diameter are in different quadrants?
4. Can the equation of a circle be expressed in parametric form? How?
5. What is the relation between the equation of a circle and its tangent lines?

**Tip:** When solving geometry problems involving circles, visualizing the circle and its key components (center, radius, etc.) can often simplify the solution process.

Find the equation of a circle in standard form where C(2, -1) and D(-1, -3) are endpoints of a diameter.

Learn how to find the equation of a circle in standard form given the endpoints of its diameter, C(2, -1) and D(-1, -3). This problem involves key geometric concepts including finding the center using the midpoint formula, calculating the radius using the distance formula, and forming the equation of a circle. Suitable for students in Grades 9-11.

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