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(−4,2) and B(−4,16).

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 based on the diameter defined by points A and B, we start with the following steps:

### Step 1: Find the midpoint (center of the circle)
The midpoint formula is given by:

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

Let \(A(-4, 2)\) and \(B(-4, 16)\):

\[
h = \frac{-4 + -4}{2} = \frac{-8}{2} = -4
\]
\[
k = \frac{2 + 16}{2} = \frac{18}{2} = 9
\]

So, the center of the circle is \((-4, 9)\).

### Step 2: Find the distance (diameter of the circle)
Using the distance formula:

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

Here, let \(x_1 = -4\), \(y_1 = 2\), \(x_2 = -4\), \(y_2 = 16\):

\[
d = \sqrt{((-4) - (-4))^2 + (16 - 2)^2}
\]

This simplifies to:

\[
d = \sqrt{(0)^2 + (14)^2} = \sqrt{196} = 14
\]

So, the measure of the diameter of the circle is \(14\).

### Step 3: Find the radius of the circle
The radius is half of the diameter:

\[
r = \frac{d}{2} = \frac{14}{2} = 7
\]

### Step 4: Write the equation of the circle
The general equation of a circle is given by:

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

Substituting \(h = -4\), \(k = 9\), and \(r = 7\) into the equation:

\[
(x - (-4))^2 + (y - 9)^2 = 7^2
\]
\[
(x + 4)^2 + (y - 9)^2 = 49
\]

### Final Answers
- **Measure of the diameter of the circle**: \(14\)
- **Measure of the radius of the circle**: \(7\)
- **Center of the circle (h, k)**: \((-4, 9)\)
- **Equation of the circle**: \((x + 4)^2 + (y - 9)^2 = 49\)