Question

Center point: (2, -2)
Perimeter point (6, -2)

Which equation is the standard form of the equation of a circle drawn on the coordinate plane, derived from the Pythagorean Theorem?

a. (x + 2)^2 + (y - 6)^2 = 4
b. (x - 2)^2 + (y - 2)^2 = 4
c. (x - 6)^2 + (y - 2)^2 = 16
d. (x - 2)^2 + (y + 2)^2 = 16

Answer 1

To find the equation of a circle in standard form, we need to 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.

Given:
- Center point: \( (2, -2) \)
- Perimeter point: \( (6, -2) \)

1. **Identify the center**: The center \( (h, k) \) is \( (2, -2) \).

2. **Calculate the radius**: The radius is the distance from the center to the perimeter point. We can find it using the distance formula (or by simply calculating the horizontal distance, since both points have the same \(y\)-coordinate):

\[
r = \sqrt{(6 - 2)^2 + (-2 - (-2))^2} = \sqrt{(4)^2 + (0)^2} = \sqrt{16} = 4
\]

Thus, \(r = 4\).

3. **Form the equation**: Substituting the center and radius into the standard form:

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

This simplifies to:

\[
(x - 2)^2 + (y + 2)^2 = 16
\]

Now we can look for the matching option among the given choices:

- a. \( (x + 2)^2 + (y - 6)^2 = 4 \) (Incorrect center and radius)
- b. \( (x - 2)^2 + (y - 2)^2 = 4 \) (Incorrect y-coordinate for center and incorrect radius)
- c. \( (x - 6)^2 + (y - 2)^2 = 16 \) (Incorrect center)
- d. \( (x - 2)^2 + (y + 2)^2 = 16 \) (Correct)

The correct answer is:

**d. \( (x - 2)^2 + (y + 2)^2 = 16 \)**