Understanding the Center and Radius of a Given Circle Equation

When working with algebraic equations that represent geometric shapes, it is crucial to accurately determine the center and radius of the shapes. This is particularly important in the context of circles. In this article, we will delve into the method to determine the center and radius of a circle from its given equation.

Given Equation and Algebraic Manipulation

Consider the given equation:

x2 y2 - 4x - 6y - 20 0

To find the center and radius of the circle, we need to rewrite the equation in the standard form of a circle's equation, which is:

(x - h)2 (y - k)2 r2, where (h, k) is the center and r is the radius.

Step-by-Step Solution

Start by grouping the x and y terms and move the constant term to the right side of the equation:

x2 - 4x y2 - 6y - 20 0

2. Complete the square for the x terms and the y terms:

x2 - 4x 4 y2 - 6y 9 - 20 - 4 - 9 0

3. Simplify:

(x - 2)2 (y - 3)2 25

4. From the standard form, we can directly read off the center and radius:

Center (2, -3)

Radius √25 5

Verification and Misconceptions

It is important to verify the steps and avoid common misconceptions. For instance, there might be confusion about the correct steps to complete the square and the correct interpretation of the equation.

Misconceptions in the Given Equations

Let us examine and correct the following common errors:

Incorrect steps in completing the square Incorrect interpretation of the standard form Incorrect calculation of the radius using the discriminant method

For example, in one of the given equations:

x2/4y2/4 - 3x - 6y - 20 0
This is not a circle equation but might represent a more complex conic section. Proper identification and transformation are necessary.

x2/4 y2/4 - 3x - 6y - 20 0
Transformation to standard form involves proper algebraic manipulation.

Conclusion

By mastering the algebraic techniques of completing the square and applying them to the given circle equation, we can accurately determine the center and radius. Understanding these concepts is crucial for solving various problems in geometry and algebra.

Key Takeaways

The standard form of a circle's equation is (x - h)2 (y - k)2 r2. Completing the square is a fundamental step in transforming the given equation. Correct interpretation of the transformed equation will lead to accurate center and radius calculations.