Find the center and radius equation X ^2 y ^2 + 12 X -6 Y -7 is equal to 0

The given equation appears to have a typo or some missing components since it is written as \(x^2y^2 + 12x - 6y - 7 = 0\), which does not correspond to the standard form of any conic section. I will assume you meant to write a standard form equation of a circle.

If the intended equation is a circle, the correct form should be:

\[
x^2 + y^2 + 12x - 6y - 7 = 0.
\]

### Step-by-Step Solution

To find the center and radius of the circle from this equation, we need to rewrite it in the standard form:

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

where \((h, k)\) is the center and \(r\) is the radius. To do this, we'll complete the square for both \(x\) and \(y\).

#### 1. Rearrange the equation

Let's group the \(x\) and \(y\) terms together:

\[
x^2 + 12x + y^2 - 6y = 7.
\]

#### 2. Complete the square for \(x\) and \(y\)

To complete the square:

- **For \(x^2 + 12x\):**
  \[
  x^2 + 12x = (x^2 + 12x + 36) - 36 = (x + 6)^2 - 36.
  \]

- **For \(y^2 - 6y\):**
  \[
  y^2 - 6y = (y^2 - 6y + 9) - 9 = (y - 3)^2 - 9.
  \]

Now substitute these back into the equation:

\[
(x + 6)^2 - 36 + (y - 3)^2 - 9 = 7.
\]

#### 3. Simplify the equation

Combine like terms:

\[
(x + 6)^2 + (y - 3)^2 - 45 = 7.
\]

Add 45 to both sides to get the equation in standard form:

\[
(x + 6)^2 + (y - 3)^2 = 52.
\]

#### 4. Identify the center and radius

The equation \((x + 6)^2 + (y - 3)^2 = 52\) is now in the standard form of a circle. From this, we can see:

- **Center:** \((h, k) = (-6, 3)\)
- **Radius:** \(r = \sqrt{52} = 2\sqrt{13}\)

### Conclusion

The center of the circle is at \((-6, 3)\) and the radius is \(2\sqrt{13}\).

Would you like further clarification on the process of completing the square or have any other questions?

### Further Related Questions

1. How do you find the equation of a circle given the endpoints of its diameter?
2. What is the process for converting a general quadratic equation into the standard form of a circle?
3. How do you determine if a point lies inside, on, or outside a given circle?
4. What is the significance of the radius in relation to a circle's curvature?
5. How do changes in the coefficients of \(x\) and \(y\) affect the position and size of a circle?

### Tip:

When dealing with conic sections, completing the square is a powerful technique to transform a general equation into a standard form, making it easier to identify key characteristics such as center, radius, foci, vertices, and asymptotes.

Learn how to find the center and radius of a circle from the equation x^2 + y^2 + 12x - 6y - 7 = 0. This problem involves completing the square technique to transform the equation into standard form, identifying key characteristics such as center and radius. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation