Question

a circle has the equation x² + y² - x + 6y + 9 = 0. (a) find the center (h,k) and radius r of the circle. (b) graph the circle. (c) find the intercepts, if any, of the graph. (a) the center of the circle is . (type an ordered pair, using integers or fractions.)

Explanation:

Step1: Rewrite the equation in standard form

The general equation of a circle is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center and $r$ is the radius. Given $x^{2}+y^{2}-x + 6y+9 = 0$. Complete the square for $x$ and $y$ terms.
For the $x$ - terms: $x^{2}-x=(x-\frac{1}{2})^{2}-\frac{1}{4}$. For the $y$ - terms: $y^{2}+6y=(y + 3)^{2}-9$.
So the equation becomes $(x-\frac{1}{2})^{2}-\frac{1}{4}+(y + 3)^{2}-9+9=0$.

Step2: Simplify the equation

$(x-\frac{1}{2})^{2}+(y + 3)^{2}=\frac{1}{4}$.

Answer:

$(\frac{1}{2},-3)$