https://file.aiquickdraw.com/mathbot/file/1724812858068vk08fvhr.jpg

https://file.aiquickdraw.com/mathbot/file/1724812858069a6z4ydkt.jpg

To determine the equation of the circle graphed in the image, we need to use the standard form of the equation of a circle, which is:

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

Here:
- \((h, k)\) is the center of the circle.
- \(r\) is the radius of the circle.

### Step 1: Identify the Center
From the image, the center of the circle is given as \((4, 6)\).

### Step 2: Determine the Radius
The radius \(r\) can be determined by using the distance between the center and any point on the circle. The point \((7, 7)\) lies on the circle, so we use the distance formula:

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

Substituting the points \((4, 6)\) and \((7, 7)\):

\[
r = \sqrt{(7 - 4)^2 + (7 - 6)^2} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}
\]

### Step 3: Write the Equation
Now that we know the center \((h, k) = (4, 6)\) and the radius \(r = \sqrt{10}\), we can write the equation of the circle:

\[
(x - 4)^2 + (y - 6)^2 = (\sqrt{10})^2
\]

Simplifying, we get:

\[
(x - 4)^2 + (y - 6)^2 = 10
\]

### Final Answer
The equation of the circle is:

\[
(x - 4)^2 + (y - 6)^2 = 10
\]

Would you like any further details or have any questions? Here are some related questions for further practice:

1. How do you derive the distance formula used to find the radius?
2. How would the equation change if the radius were doubled?
3. If the center was moved to \((2, -3)\), what would be the new equation of the circle?
4. How can you verify if a point lies inside, on, or outside a circle given its equation?
5. How would you find the intercepts of the circle with the x-axis and y-axis?

**Tip:** Understanding the components of the circle's equation helps in analyzing and graphing circles quickly.

Learn how to find the equation of a circle given its center at (4, 6) and radius sqrt(10). This problem covers circle geometry, including the equation of a circle and the distance formula used to determine the radius. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation