Question

An artist is creating a decorative round mirror and needs to find the y-coordinate of the mirror's center to ensure that it is vertically aligned. The mirror's edge is defined by the equation x2+y2+4x−6y=0 . What is the y-coordinate?(

Answer 1

To find the y-coordinate of the center of the mirror defined by the equation \(x^2 + y^2 + 4x - 6y = 0\), we start by rewriting the equation in a more familiar form by completing the square.

1. **Reorganize the equation:**

\[
x^2 + 4x + y^2 - 6y = 0
\]

2. **Complete the square for \(x\):**

Take the \(x\) terms \(x^2 + 4x\):
\[
x^2 + 4x \rightarrow (x + 2)^2 - 4
\]

3. **Complete the square for \(y\):**

Take the \(y\) terms \(y^2 - 6y\):
\[
y^2 - 6y \rightarrow (y - 3)^2 - 9
\]

4. **Substitute back into the equation:**

Now substituting these completed squares back into the equation, we have:
\[
(x + 2)^2 - 4 + (y - 3)^2 - 9 = 0
\]

Simplifying this gives:
\[
(x + 2)^2 + (y - 3)^2 - 13 = 0
\]

Rearranging the equation:
\[
(x + 2)^2 + (y - 3)^2 = 13
\]

5. **Identify the center of the circle:**

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

From our equation \((x + 2)^2 + (y - 3)^2 = 13\), we can see that:
- The center \((h, k)\) is \((-2, 3)\).

6. **Find the y-coordinate of the center:**

Therefore, the y-coordinate of the mirror's center is:
\[
\boxed{3}
\]