Question

A circle is drawn on a coordinate 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 standard equation of a circle in the coordinate plane is given by:

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

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

Given the center at \((-4, 6)\) and a radius of 2, we can substitute these values into the equation:

- \(h = -4\)
- \(k = 6\)
- \(r = 2\), so \(r^2 = 2^2 = 4\)

Thus, 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 can fill in the missing information in the provided format \( (x - 1.___)^2 + (y - 2.___)^2 = 3.___ \).

1. For \(x + 4\), we need the term \(x - 1.___\) to equal \(x + 4\):
\[
1.___ = -4 \quad \Rightarrow \quad \text{So the answer is } \text{a. } -4
\]

2. For \(y - 6\), we need the term \(y - 2.___\) to equal \(y - 6\):
\[
2.___ = 6 \quad \Rightarrow \quad \text{So the answer is } \text{a. } 6
\]

3. For \(r^2\), the equation has \(4\) on the right side:
\[
3.___ = 4 \quad \Rightarrow \quad \text{So the answer is } \text{b. } 4
\]

Putting it together, the answers to the questions are:

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