The problem asks us to analyze the equation $x^2 + y^2 - 8x + 4y + 13 = 0$. We need to determine what this equation represents. We can do this by completing the square for both $x$ and $y$ terms.

1. Problem Description

The problem asks us to analyze the equation $x^2 + y^2 - 8x + 4y + 13 = 0$. We need to determine what this equation represents. We can do this by completing the square for both $x$ and $y$ terms.

2. Solution Steps

We are given the equation $x^2 + y^2 - 8x + 4y + 13 = 0$.

We group the $x$ terms and the $y$ terms together:

$(x^2 - 8x) + (y^2 + 4y) + 13 = 0$.

Now we complete the square for the $x$ terms. We take half of the coefficient of the $x$ term, which is $-8/2 = -4$, and square it: $(-4)^2 = 16$. We add and subtract

1

6.

$x^2 - 8x + 16 - 16 = (x-4)^2 - 16$.

Next, we complete the square for the $y$ terms. We take half of the coefficient of the $y$ term, which is $4/2 = 2$, and square it: $(2)^2 = 4$. We add and subtract

4.

$y^2 + 4y + 4 - 4 = (y+2)^2 - 4$.

Now we substitute these back into the original equation:

$(x-4)^2 - 16 + (y+2)^2 - 4 + 13 = 0$.

$(x-4)^2 + (y+2)^2 - 16 - 4 + 13 = 0$.

$(x-4)^2 + (y+2)^2 - 7 = 0$.

$(x-4)^2 + (y+2)^2 = 7$.

This is the equation of a circle with center $(4, -2)$ and radius $\sqrt{7}$.

3. Final Answer

The equation represents a circle with center $(4, -2)$ and radius $\sqrt{7}$.