Question

The diameter of a circle is shown. The coordinates are (-8,4) and (-4,-2) Fill in the missing value in the equation of the circle.(1 point)(x+6)2+(y−1)2=

Answer 1

To find the missing value in the equation of the circle, we first need to determine the center (h, k) and the radius of the circle from the given points on the diameter.

The endpoints of the diameter are given as (-8, 4) and (-4, -2).

1. **Finding the center of the circle:**
The center (h, k) can be calculated as the midpoint of the endpoints of the diameter:

\[
h = \frac{x_1 + x_2}{2} = \frac{-8 + (-4)}{2} = \frac{-12}{2} = -6
\]

\[
k = \frac{y_1 + y_2}{2} = \frac{4 + (-2)}{2} = \frac{2}{2} = 1
\]

So, the center of the circle is at (-6, 1).

2. **Finding the radius of the circle:**
The radius can be calculated by finding the distance from the center to one of the endpoints (let's use (-8, 4)).

The distance formula is given by:

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

Substituting the center (-6, 1) and the point (-8, 4):

\[
r = \sqrt{(-8 - (-6))^2 + (4 - 1)^2} = \sqrt{(-8 + 6)^2 + (4 - 1)^2} = \sqrt{(-2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13}
\]

3. **Writing the equation of the circle:**
The standard equation of a circle with center (h, k) and radius r is:

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

We already found that (h, k) = (-6, 1) and \( r^2 = 13 \).

Thus, the equation of the circle is:

\[
(x + 6)^2 + (y - 1)^2 = 13
\]

So, the missing value in the equation of the circle is:

\[
\boxed{13}
\]