Question

circle is drawn on a coordinate plane with the center point at (−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.(2 points)
(x−
)2+
(y−
)2=

Answer 1

The standard equation of a circle with center \((h, k)\) and radius \(r\) is given by:

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

For the given circle, the center is at \((-4, 6)\) and the radius is \(2\).

Here, \(h = -4\), \(k = 6\), and \(r = 2\).

Now we can substitute these values into the standard equation:

1. Substitute for \(h\):
- Include \(-4\) in the equation: \((x - (-4))\)

2. Substitute for \(k\):
- Include \(6\) in the equation: \((y - 6)\)

3. Substitute for \(r\):
- The radius \(r = 2\), therefore \(r^2 = 2^2 = 4\).

Putting these values together results in the equation of the circle:

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

So, filling in the missing information in your provided equation format, we get:

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

Thus, the completed equations are:

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

This indicates:
- The first blank is \(-4\)
- The second blank is \(6\)
- The third blank is \(4\)