Question
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.)
Ask by Martinez Hardy. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
- **Center:** \((0, 0)\)
- **Radius:** \(8\)
- **x-intercepts:** \((8, 0)\) and \((-8, 0)\)
- **y-intercepts:** \((0, 8)\) and \((0, -8)\)
Solution
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:** \((0, 8)\), \((0, -8)\)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
The center of the circle given the equation \( x^{2}+y^{2}=64 \) is at the point (0, 0), which is the origin. The radius of the circle is calculated by taking the square root of 64, giving us a radius of 8. So, the ordered pair for the center is (0, 0) and the radius is 8. When graphing the circle, you'd start at the center (0, 0) and mark points that are 8 units away in all directions—up (0, 8), down (0, -8), left (-8, 0), and right (8, 0). Then, you can sketch the circle by connecting these points smoothly. This will yield a perfect circle that intersects the X-axis at (-8, 0) and (8, 0), and the Y-axis at (0, -8) and (0, 8).
