\begin{tabular}{l|l|} estion list & \multicolumn{1}{l}{$$ \begin{array}{l} ext { A circle has the equation } x^{2}+y^{2}-3 x-6 y+9=0 . \ ext { (a) Find the center (h,k) and radius r of the circle. } \ ext { (b) Graph the circle. } \ ext { (c) Find the intercepts, if any, of the graph. }\end{array} $$} \ Question 7 & $$ \begin{array}{l} ext { (a) The center of the circle is }\left(\frac{3}{2}, 3 ight) . \ ext { (Type an ordered pair, using integers or fractions.) } \ ext { Question } 8 \ ext { Que radius of the circle is unknown. } \ ext { (Type an integer or a fraction.) }\end{array} $$ \ \hline\end{tabular}

Question

\begin{tabular}{l|l|} estion list & \multicolumn{1}{l}{$$ \begin{array}{l}	ext { A circle has the equation } x^{2}+y^{2}-3 x-6 y+9=0 . \ 	ext { (a) Find the center (h,k) and radius r of the circle. } \ 	ext { (b) Graph the circle. } \ 	ext { (c) Find the intercepts, if any, of the graph. }\end{array} $$} \ Question 7 & $$ \begin{array}{l}	ext { (a) The center of the circle is }\left(\frac{3}{2}, 3ight) . \ 	ext { (Type an ordered pair, using integers or fractions.) } \ 	ext { Question } 8 \ 	ext { Que radius of the circle is unknown. } \ 	ext { (Type an integer or a fraction.) }\end{array} $$ \ \hline\end{tabular}

UpStudy AI · Accepted Answer

**Question 8**

**Find the radius of the circle.**

Given the equation of the circle:
$$
x^{2} + y^{2} - 3x - 6y + 9 = 0
$$

**Step 1: Rewrite the equation by completing the square for both $$ x $$ and $$ y $$.**

$$
x^{2} - 3x + y^{2} - 6y = -9
$$

**Complete the square for $$ x $$:**
$$
x^{2} - 3x = \left(x - \frac{3}{2}\right)^{2} - \left(\frac{3}{2}\right)^{2} = \left(x - \frac{3}{2}\right)^{2} - \frac{9}{4}
$$

**Complete the square for $$ y $$:**
$$
y^{2} - 6y = (y - 3)^{2} - 9
$$

**Substitute back into the equation:**
$$
\left(x - \frac{3}{2}\right)^{2} - \frac{9}{4} + (y - 3)^{2} - 9 = -9
$$

**Simplify:**
$$
\left(x - \frac{3}{2}\right)^{2} + (y - 3)^{2} = \frac{9}{4}
$$

**Step 2: Identify the radius.**

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

From the simplified equation:
$$
\left(x - \frac{3}{2}\right)^{2} + (y - 3)^{2} = \frac{9}{4}
$$
we can see that:
$$
r^{2} = \frac{9}{4}
$$
$$
r = \sqrt{\frac{9}{4}} = \frac{3}{2}
$$

**Answer:** The radius of the circle is $$ \frac{3}{2} $$.
The radius of the circle is $$ \frac{3}{2} $$.

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