Question

for the equation x² + y² - 6x - 2y - 6 = 0, do the following. (a) find the center (h,k) and radius r of the circle. (b) graph the circle. (c) find the intercepts, if any. (a) the center is . (type an ordered pair.)

Explanation:

Step1: Rewrite the given 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 $x^{2}+y^{2}-6x - 2y-6 = 0$. Complete the square for $x$ and $y$ terms.
For the $x$ - terms: $x^{2}-6x=(x - 3)^{2}-9$.
For the $y$ - terms: $y^{2}-2y=(y - 1)^{2}-1$.
So the equation becomes $(x - 3)^{2}-9+(y - 1)^{2}-1-6 = 0$.

Step2: Simplify the equation

$(x - 3)^{2}+(y - 1)^{2}=9 + 1+6$.
$(x - 3)^{2}+(y - 1)^{2}=16$.

Answer:

$(3,1)$