Find the centre and radius of the circle  x²-8x+y²+8y=4

To find the center and radius of the circle described by the equation:

\[
x^2 - 8x + y^2 + 8y = 4,
\]

we'll rewrite the equation in standard form. The general 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.

### Steps:

1. **Group the \(x\)-terms and \(y\)-terms**:
   \[
   (x^2 - 8x) + (y^2 + 8y) = 4.
   \]

2. **Complete the square for the \(x\)-terms**:
   - Take half of \(-8\), square it, and add it to both sides: 
     \[
     \left(\frac{-8}{2}\right)^2 = 16.
     \]
   - Add \(16\) to both sides:
     \[
     (x^2 - 8x + 16).
     \]

3. **Complete the square for the \(y\)-terms**:
   - Take half of \(8\), square it, and add it to both sides:
     \[
     \left(\frac{8}{2}\right)^2 = 16.
     \]
   - Add \(16\) to both sides:
     \[
     (y^2 + 8y + 16).
     \]

4. **Rewrite the equation**:
   Now the equation looks like this:
   \[
   (x^2 - 8x + 16) + (y^2 + 8y + 16) = 4 + 16 + 16,
   \]
   which simplifies to:
   \[
   (x - 4)^2 + (y + 4)^2 = 36.
   \]

5. **Interpret the equation**:
   The equation is now in standard form:
   \[
   (x - 4)^2 + (y + 4)^2 = 6^2.
   \]

### Conclusion:

- The **center** of the circle is \((4, -4)\).
- The **radius** of the circle is \(6\).

Would you like more details or have any questions?

---

Here are 5 related questions for further practice:

1. How would you complete the square for a different circle equation, such as \(x^2 + 6x + y^2 - 4y = 5\)?
2. How do you find the equation of a circle given the center and a point on the circle?
3. Can you find the distance between the center of the circle \((4, -4)\) and another point \((1, 2)\)?
4. What is the general equation of a circle that passes through the origin?
5. How would you calculate the area of a circle given its radius?

**Tip:** Completing the square is a useful method not just for circles, but also in solving quadratic equations and finding the vertex of a parabola!

Find the centre and radius of the circle x²-8x+y²+8y=4

Learn how to find the center and radius of the circle given by the equation x²-8x+y²+8y=4. This solution includes steps to complete the square, transform the equation into standard form, and identify the circle's 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