Question

A circle is drawn on a coordinate plane with the center point at (−4,6)(−4,6) and a radius of 2. Derive the equation of the circle from the given information. Fill in the missing information in the following equation.(3 points)
(x-
)2
+(y-
)2
=

Answer 1

To derive the equation of a circle given its center and radius, we use the standard form of the 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 given information:
- The center of the circle \((h, k)\) is \((-4, 6)\).
- The radius \(r\) is \(2\).

We substitute these values into the standard form:

1. \(h = -4\)
2. \(k = 6\)
3. \(r = 2\) so \(r^2 = 2^2 = 4\)

Putting this into the equation, we get:

\[
(x - (-4))^2 + (y - 6)^2 = 4
\]

This simplifies to:

\[
(x + 4)^2 + (y - 6)^2 = 4
\]

Now filling in the missing information in the format you provided:

- \(x - \) value: -4
- \(y - \) value: 6
- Right side of the equation: 4

Thus, the filled equation is:

\[
(x + 4)^2 + (y - 6)^2 = 4
\]
with the missing information completed as follows:
\[
(x - (-4))^2 + (y - 6)^2 = 4
\]

So, the missing components are:
- \( -4 \)
- \( 6 \)
- \( 4 \)