A circle has the equation 2x squaredplus20xplus2y squaredequals0.
​(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.
Question content area bottom left
Part 1
​(a) The center of the circle is

enter your response here.
​(Type an ordered​ pair, using integers or​ decimals.)
Part 2
The radius of the circle is

enter your response here.
​(Type an integer or a​ decimal.)
Part 3
​(b) Use the graphing tool to graph the circle.

Part 4
​(c) What are the​ intercepts? Select the correct choice below​ and, if​ necessary, fill in the answer box within your choice.
A.
The​ intercept(s) of the circle​ is/are

enter your response here.
​(Type an ordered pair. Use a comma to separate answers as needed. Type exact answers for each​ coordinate, using radicals as​ needed.)
B.
There are no intercepts.

Let's go through the problem step by step.

### **Given:**
The equation of the circle is:
\[ 2x^2 + 20x + 2y^2 = 0 \]

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

1. **Simplify the equation** by dividing the entire equation by 2:
   \[
   x^2 + 10x + y^2 = 0
   \]

2. **Complete the square** for the \(x\) terms:
   - Take the coefficient of \(x\), which is 10, halve it to get 5, and then square it to get 25.
   - Add and subtract 25 inside the equation:
   \[
   (x^2 + 10x + 25) + y^2 - 25 = 0
   \]
   - Rewrite the equation:
   \[
   (x + 5)^2 + y^2 = 25
   \]

This equation is now in the standard form of a circle:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center and \(r\) is the radius.

- **Center** \((h, k)\) is \((-5, 0)\).
- **Radius** \(r\) is \(5\) (since \(r^2 = 25\)).

So:
- **Center** \((h, k)\) = \((-5, 0)\)
- **Radius** \(r = 5\)

### **(b) Graph the circle.**

To graph the circle:
- Plot the center at \((-5, 0)\).
- Use the radius \(5\) to mark points 5 units away from the center in all directions (left, right, up, and down).

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

1. **X-Intercepts:**
   - Set \(y = 0\) in the equation:
   \[
   (x + 5)^2 = 25
   \]
   \[
   x + 5 = \pm 5
   \]
   \[
   x = 0 \text{ or } x = -10
   \]
   So, the x-intercepts are \((0, 0)\) and \((-10, 0)\).

2. **Y-Intercepts:**
   - Set \(x = 0\) in the equation:
   \[
   (0 + 5)^2 + y^2 = 25
   \]
   \[
   25 + y^2 = 25
   \]
   \[
   y^2 = 0 \quad \Rightarrow \quad y = 0
   \]
   So, the y-intercept is \((0, 0)\).

### **Summary of Results:**

- **Part 1: Center:** \((-5, 0)\)
- **Part 2: Radius:** \(5\)
- **Part 4: Intercepts:** The circle intercepts the axes at \((0, 0)\) and \((-10, 0)\).

### Would you like a detailed explanation of any part? Here are 5 related questions to consider:

1. How do you complete the square for equations involving both \(x\) and \(y\) terms?
2. What is the general form of a circle's equation and how does it relate to the standard form?
3. How do you determine intercepts for any general circle equation?
4. What methods can be used to graph a circle given its equation?
5. How does the radius of a circle affect its position relative to the axes?

**Tip:** Always simplify the equation of a circle to its standard form to easily identify its center and radius.

Learn how to solve the circle equation 2x^2 + 20x + 2y^2 = 0 step-by-step. Find the center (h, k) and radius r of the circle, graph the circle, and determine its intercepts. This problem covers circle equations, completing the square method, graphing circles, and finding intercepts. Suitable for high school students.

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