Question

a circle is drawn on a coordinate plane with the center point at (-8, -2) and a radius of 2. derive the equation of the circle in standard form, using the given information and the pythagorean theorem.
a. (x+2)^2 + (y+8)^2 = 2
b. (x - 2)^2 + (y - 8)^2 = 2
c. (x - 8)^2 + (y - 2)^2 = 4
d. (x+8)^2 + (y + 2)^2 = 4

Answer 1

To derive the equation of a circle in standard form given the center and the radius, we use the standard form equation of a circle:

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

where \((h, k)\) is the center of the circle and \(r\) is the radius.

From the information provided:
- The center of the circle is at \((-8, -2)\), so \(h = -8\) and \(k = -2\).
- The radius \(r = 2\).

Now calculating \(r^2\):
\[
r^2 = 2^2 = 4
\]

Substituting \(h\), \(k\), and \(r^2\) into the standard form equation:

\[
(x - (-8))^2 + (y - (-2))^2 = 4
\]
This simplifies to:
\[
(x + 8)^2 + (y + 2)^2 = 4
\]

Thus, the equation of the circle in standard form is:

\[
(x + 8)^2 + (y + 2)^2 = 4
\]

Now, looking at the choices:
- a. \((x + 2)^2 + (y + 8)^2 = 2\)
- b. \((x - 2)^2 + (y - 8)^2 = 2\)
- c. \((x - 8)^2 + (y - 2)^2 = 4\)
- d. \((x + 8)^2 + (y + 2)^2 = 4\)

The correct answer is:
**d. \((x + 8)^2 + (y + 2)^2 = 4\)**