Question

An artist is creating a circular mosaic and needs to determine the x-coordinate of the center for proper layout. The boundary of the mosaic is described by the equation x2+y2+4x−6y=0
. What is the x-coordinate?(1 point)
The x-coordinate is
.

Answer 1

To find the x-coordinate of the center of the circular mosaic given by the equation \( x^2 + y^2 + 4x - 6y = 0 \), we first need to rewrite this equation in standard form by completing the square.

1. Start with the given equation:
\[
x^2 + y^2 + 4x - 6y = 0
\]

2. Rearrange the equation:
\[
x^2 + 4x + y^2 - 6y = 0
\]

3. Complete the square for \( x \):
- For \( x^2 + 4x \), take half of 4, square it, giving \( 4 \):
\[
x^2 + 4x = (x + 2)^2 - 4
\]

4. Complete the square for \( y \):
- For \( y^2 - 6y \), take half of -6, square it, giving \( 9 \):
\[
y^2 - 6y = (y - 3)^2 - 9
\]

5. Substitute these back into the equation:
\[
(x + 2)^2 - 4 + (y - 3)^2 - 9 = 0
\]
\[
(x + 2)^2 + (y - 3)^2 - 13 = 0
\]
\[
(x + 2)^2 + (y - 3)^2 = 13
\]

Now we have the equation in standard form, which describes a circle:
\[
(x + 2)^2 + (y - 3)^2 = 13
\]
From this equation, we can identify the center of the circle at the coordinates \( (-2, 3) \).

Therefore, the x-coordinate of the center is:
\[
\text{x-coordinate} = -2
\]

So the answer is \(-2\).