$$ \leftarrow $$ A circle has the equation $$ x^{2}+y^{2}-3 x-6 y+9=0 $$. (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.)

Question

UpStudy AI · Accepted Answer

To find the center $$(h, k)$$ and the radius $$r$$ of the circle given by the equation:

$$
x^{2} + y^{2} - 3x - 6y + 9 = 0
$$

we'll complete the square for both $$x$$ and $$y$$.

### Step 1: Rearrange the Equation

Group the $$x$$ terms and the $$y$$ terms:

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

### Step 2: Complete the Square

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

**For $$y$$:**
$$
y^{2} - 6y \
= y^{2} - 6y + \left(\frac{6}{2}\right)^2 - \left(\frac{6}{2}\right)^2 \
= (y - 3)^2 - 9
$$

### Step 3: Substitute Back into the Equation

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

Combine like terms:

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

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

### Step 4: Identify the Center and Radius

The equation is now in standard form:

$$
\left(x - h\right)^2 + \left(y - k\right)^2 = r^2
$$

Comparing, we find:

- **Center $$(h, k)$$:** $$\left(\frac{3}{2},\ 3\right)$$
- **Radius $$r$$:** $$\sqrt{\frac{9}{4}} = \frac{3}{2}$$

---

**Answer to Part (a):**

The center of the circle is $$\left(\dfrac{3}{2},\ 3\right)$$.
The center of the circle is $$\left(\dfrac{3}{2},\ 3\right)$$.

A circle has the equation .
(a) Find the center and radius r of the circle.
(b) Graph the circle.
© Find the intercepts, if any, of the graph.
(a) The center of the circle is
(Type an ordered pair, using integers or fractions.)

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Solución

Step 1: Rearrange the Equation

Step 2: Complete the Square

Step 3: Substitute Back into the Equation

Step 4: Identify the Center and Radius

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A circle has the equation . (a) Find the center and radius r of the circle. (b) Graph the circle. © Find the intercepts, if any, of the graph. (a) The center of the circle is (Type an ordered pair, using integers or fractions.)