Question

Use the following formulas to help you find the equation of a circle give two points that are on the diameter of the circle.
distance: d=(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√
midpoint: (x,y)=(x1+x22,y1+y22)
circle: (x−h)2+(y−k)2=r2(5 points)
Write the equation for a circle with a diameter AB¯¯¯¯¯¯¯¯ given coordinates A(−8,−8) and B(−2,−8).

Measure of the diameter of the circle:

Measure of the radius of the circle:

Center of the circle (h,k): (
,
)

Write the entire equation of the circle:

Answer 1

To find the equation of a circle given two points A and B that are on the diameter, we will apply the distance formula to find the diameter, the midpoint formula for the center of the circle, and finally use the standard equation of a circle.

Given points:
- A(-8, -8)
- B(-2, -8)

### Step 1: Find the distance (diameter)

Using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting coordinates of points A and B:
- \(x_1 = -8\)
- \(y_1 = -8\)
- \(x_2 = -2\)
- \(y_2 = -8\)

Calculating:
\[
d = \sqrt{((-2) - (-8))^2 + ((-8) - (-8))^2}
\]
\[
= \sqrt{(-2 + 8)^2 + (0)^2}
\]
\[
= \sqrt{(6)^2 + 0}
\]
\[
= \sqrt{36}
\]
\[
= 6
\]

Thus, the **measure of the diameter** of the circle is **6**.

### Step 2: Calculate the radius

The radius \(r\) is half the diameter:
\[
r = \frac{d}{2} = \frac{6}{2} = 3
\]

Thus, the **measure of the radius** of the circle is **3**.

### Step 3: Find the center of the circle (h, k)

Using the midpoint formula:
\[
(h, k) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]

Substituting the coordinates of points A and B:
\[
(h, k) = \left(\frac{-8 + (-2)}{2}, \frac{-8 + (-8)}{2}\right)
\]
\[
= \left(\frac{-10}{2}, \frac{-16}{2}\right)
\]
\[
= (-5, -8)
\]

Thus, the **center of the circle** is \((-5, -8)\).

### Step 4: Write the entire equation of the circle

Using the standard equation of a circle:
\[
(x - h)^2 + (y - k)^2 = r^2
\]

Substituting \(h = -5\), \(k = -8\), and \(r = 3\):
\[
(x - (-5))^2 + (y - (-8))^2 = 3^2
\]
which simplifies to:
\[
(x + 5)^2 + (y + 8)^2 = 9
\]

### Final Outputs
- Measure of the diameter of the circle: **6**
- Measure of the radius of the circle: **3**
- Center of the circle (h, k): **(-5, -8)**
- Equation of the circle: **\((x + 5)^2 + (y + 8)^2 = 9\)**