The endpoints of the diameter of circle are (0,6) and (0,-4) what is the equation of the circle?

To find the equation of a circle, we need the **center** \((h, k)\) and the **radius** \(r\). The general form of the equation of a circle is:

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

### Step 1: Find the center of the circle
The center is the midpoint of the diameter. The endpoints of the diameter are \((0, 6)\) and \((0, -4)\). The midpoint formula is:

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

Substitute the values:

\[
\left( \frac{0 + 0}{2}, \frac{6 + (-4)}{2} \right) = (0, 1)
\]

So, the center of the circle is \((0, 1)\).

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

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

Substitute the values:

\[
d = \sqrt{(0 - 0)^2 + (-4 - 6)^2} = \sqrt{0 + (-10)^2} = \sqrt{100} = 10
\]

The radius \(r\) is half of the diameter:

\[
r = \frac{10}{2} = 5
\]

### Step 3: Write the equation of the circle
Now that we have the center \((0, 1)\) and the radius \(r = 5\), we can substitute them into the equation of a circle:

\[
(x - 0)^2 + (y - 1)^2 = 5^2
\]

Simplifying:

\[
x^2 + (y - 1)^2 = 25
\]

Thus, the equation of the circle is:

\[
x^2 + (y - 1)^2 = 25
\]

Would you like more details or further clarification?

Here are 5 related questions you might find interesting:
1. How do you find the equation of a circle given the center and a point on the circle?
2. How can you determine the equation of a circle from its general form?
3. How would you calculate the length of a chord in a circle?
4. How does the equation of a circle change if it is shifted horizontally or vertically?
5. How do you find the area and circumference of a circle given its equation?

**Tip:** When dealing with equations of circles, always check whether the center is at the origin \((0, 0)\) or at some other point \((h, k)\), as this changes the form of the equation slightly.

The endpoints of the diameter of circle are (0,6) and (0,-4). What is the equation of the circle?

Learn how to find the equation of a circle when given the endpoints of its diameter at (0,6) and (0,-4). This step-by-step solution covers the midpoint and distance formulas, and how to apply them to determine the center and radius. Suitable for students in Grades 9-10.

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