https://file.aiquickdraw.com/mathbot/file/1729327625566xc7dd916.jpg

The problem in the image asks to:

1. **Determine the general and standard form of the circle having a diameter with endpoints A(4,7) and B(0,1).**
2. **Graph the circle.**

Let's solve it step by step:

### Step 1: Find the center of the circle
The center of a circle with a given diameter is the midpoint of the diameter.

The midpoint \( M \) of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is calculated as:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Here, the endpoints are \( A(4, 7) \) and \( B(0, 1) \).

\[
M = \left( \frac{4 + 0}{2}, \frac{7 + 1}{2} \right) = (2, 4)
\]

Thus, the center of the circle is at \( (2, 4) \).

### Step 2: Find the radius of the circle
The radius of the circle is half the length of the diameter. The length of the diameter is the distance between the points \( A(4, 7) \) and \( B(0, 1) \), calculated using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
\[
d = \sqrt{(0 - 4)^2 + (1 - 7)^2} = \sqrt{16 + 36} = \sqrt{52}
\]

The radius is half of this distance:
\[
r = \frac{\sqrt{52}}{2} = \frac{2\sqrt{13}}{2} = \sqrt{13}
\]

### Step 3: Write the standard form of the circle
The standard form of a circle's equation is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \( (h, k) \) is the center of the circle and \( r \) is the radius.

Substituting the center \( (2, 4) \) and the radius \( \sqrt{13} \):
\[
(x - 2)^2 + (y - 4)^2 = (\sqrt{13})^2
\]
\[
(x - 2)^2 + (y - 4)^2 = 13
\]

This is the **standard form** of the circle's equation.

### Step 4: Write the general form of the circle
To convert the standard form to the general form, expand the equation:
\[
(x - 2)^2 + (y - 4)^2 = 13
\]
\[
(x^2 - 4x + 4) + (y^2 - 8y + 16) = 13
\]
\[
x^2 + y^2 - 4x - 8y + 20 = 13
\]
\[
x^2 + y^2 - 4x - 8y + 7 = 0
\]

This is the **general form** of the circle's equation.

### Step 5: Graph the circle
The circle has a center at \( (2, 4) \) and a radius of \( \sqrt{13} \), which is approximately 3.61. To graph it, plot the center and draw a circle with this radius.

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Would you like more details on any step, or do you have any questions?

Here are 5 related questions to consider:
1. How is the distance formula derived?
2. How can you find the equation of a circle given three points?
3. What is the difference between the general and standard form of a circle?
4. How would the equation change if the radius were doubled?
5. How can you verify the equation of a circle if only the endpoints of the diameter are given?

**Tip**: Always double-check the radius when using the distance formula, as small errors can lead to incorrect equations.

Determine the general and standard form of the circle having a diameter with endpoints A(4,7) and B(0,1). Then graph the circle.

Solve the circle equation problem where you need to determine the general and standard form of the circle with diameter endpoints A(4,7) and B(0,1). Includes step-by-step calculation of the center, radius, and graphing the circle. Suitable for Grades 9-10 students.

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