A circle has the equation $$ x^{2}+y^{2}=64 $$. (a) Find the center (h,k) and radius rof the circle. (b) Graph the circle. (c) Find the intercepts, if any, of the graph. (a) The center of the circle is ( $$ 0 . D) $$ ). (Type an ordered pair, using integers or decimals.) The radius of the circle is $$ \square $$. (Type an integer or a decimal.)

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Let's solve each part of the problem step by step.

## Problem

A circle has the equation $$ x^{2} + y^{2} = 64 $$.

### (a) Find the center $$(h, k)$$ and radius $$ r $$ of the circle.

**Solution:**

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

Given the equation:
$$
x^{2} + y^{2} = 64
$$

We can rewrite it to match the standard form:
$$
(x - 0)^2 + (y - 0)^2 = 8^2
$$

From this comparison:
- $$ h = 0 $$
- $$ k = 0 $$
- $$ r^2 = 64 $$ $$\Rightarrow$$ $$ r = 8 $$

**Answer for (a):**
- **Center:** $$(0, 0)$$
- **Radius:** $$8$$

---

### (b) Graph the circle.

**Solution:**

To graph the circle $$ x^{2} + y^{2} = 64 $$:

1. **Center:** Plot the center at $$(0, 0)$$.
2. **Radius:** From the center, measure 8 units in all directions (up, down, left, right) to mark points at $$(8, 0)$$, $$(-8, 0)$$, $$(0, 8)$$, and $$(0, -8)$$.
3. **Draw the Circle:** Connect these points smoothly to form the circle.

*Since I can't provide a visual graph here, you can plot these points on graph paper or using a graphing tool to see the circle.*

---

### (c) Find the intercepts, if any, of the graph.

**Solution:**

**x-intercepts:**
To find the x-intercepts, set $$ y = 0 $$ and solve for $$ x $$:
$$
x^{2} + 0^{2} = 64 \
x^{2} = 64 \
x = \pm8
$$
So, the x-intercepts are at $$(8, 0)$$ and $$(-8, 0)$$.

**y-intercepts:**
To find the y-intercepts, set $$ x = 0 $$ and solve for $$ y $$:
$$
0^{2} + y^{2} = 64 \
y^{2} = 64 \
y = \pm8
$$
So, the y-intercepts are at $$(0, 8)$$ and $$(0, -8)$$.

**Answer for (c):**
- **x-intercepts:** $$(8, 0)$$ and $$(-8, 0)$$
- **y-intercepts:** $$(0, 8)$$ and $$(0, -8)$$

---

## Summary of Answers

### (a)
- **Center:** $$(0, 0)$$
- **Radius:** $$8$$

### (b)
- **Graph:** Center at $$(0, 0)$$ with a radius of 8 units.

### (c)
- **Intercepts:**
  - **x-intercepts:** $$(8, 0)$$, $$(-8, 0)$$
  - **y-intercepts:** $$(0, 8)$$, $$(0, -8)$$
- **Center:** $$(0, 0)$$ - **Radius:** $$8$$ - **x-intercepts:** $$(8, 0)$$ and $$(-8, 0)$$ - **y-intercepts:** $$(0, 8)$$ and $$(0, -8)$$

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