https://file.aiquickdraw.com/mathbot/file/172665292782590u86pza.jpg

Let's analyze the equation of the circle given in the image:

\[
x^2 + y^2 - 6x + 8y - 11 = 0
\]

We need to find the center and radius of this circle. The general form of the equation of a circle is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Where:
- \((h, k)\) is the center of the circle.
- \(r\) is the radius.

To find the center and radius, we can complete the square for both the \(x\) and \(y\) terms.

### Completing the square:

1. Group the \(x\) terms and \(y\) terms:
   \[
   (x^2 - 6x) + (y^2 + 8y) = 11
   \]

2. Complete the square for the \(x\) terms:
   \[
   x^2 - 6x = (x - 3)^2 - 9
   \]

3. Complete the square for the \(y\) terms:
   \[
   y^2 + 8y = (y + 4)^2 - 16
   \]

Now substitute these completed squares back into the equation:

\[
(x - 3)^2 - 9 + (y + 4)^2 - 16 = 11
\]

Simplify:

\[
(x - 3)^2 + (y + 4)^2 - 25 = 11
\]

Add 25 to both sides:

\[
(x - 3)^2 + (y + 4)^2 = 36
\]

Now, we can identify the center and radius:
- The center is \((3, -4)\).
- The radius is \(\sqrt{36} = 6\).

### Conclusion:
- The statement in the question says the center is \((-6, 8)\), which is **FALSE**.
- The statement says the radius is 6 units, which is **TRUE**.

Would you like any further clarifications?

---

Here are 5 follow-up questions:
1. How do you derive the general form of a circle from the given equation?
2. Can completing the square method be used for other conic sections?
3. What is the significance of the radius in a circle's equation?
4. How would you approach the problem if the equation was an ellipse instead of a circle?
5. Why is it important to simplify equations before analyzing their geometric properties?

**Tip:** Always check for terms that need completing the square in quadratic equations when dealing with circles or other conic sections.

For the circle x^2 + y^2 - 6x + 8y - 11 = 0, the centre is (-6, 8) and the radius is 6 units. Determine if the statements are TRUE or FALSE.

Analyze the circle equation x^2 + y^2 - 6x + 8y - 11 = 0 using the method of completing the square to determine the correct center and radius. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation