Question

A 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.

(x - 1.__)^2 + (y - 2.__)^2 = 3.__

1.
a. -4
b. -2
c. 4
d. 2

2.
a. 6
b. -6
c. 3
d. -3

3.
a. 4
b. -4
c. 1
d. -1

Answer 1

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

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

From the information provided, the center of the circle is at \((-4, 6)\) and the radius is \(2\). Thus:

- \(h = -4\)
- \(k = 6\)
- \(r = 2\)

Now we can plug these values into the equation:

The radius squared is:
\[
r^2 = 2^2 = 4
\]

Therefore, the equation of the circle becomes:

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

This simplifies to:

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

Now, we'll fill in the missing information in the provided equation format \((x - 1.__)^2 + (y - 2.__)^2 = 3.__\):

1. For \(x\), the value is \(+4\), which corresponds to \(-(-4)\). Hence, \(1.__ = 4\), and we select:
- a. **-4**

2. For \(y\), we have \(y - 6\), so \(2.__ = 6\), and we select:
- a. **6**

3. For the right side, we have \(4\) from \(r^2\), thus \(3.__ = 4\), and we select:
- a. **4**

So the answers are:

1. **a. -4**
2. **a. 6**
3. **a. 4**