Determining the Center and Radius of a Circle from its Equation

Understanding the standard equation of a circle is crucial in solving problems related to circles in geometry and algebra. The standard equation for a circle with center at (a, b) and radius r is:

The Standard Equation of a Circle

The standard equation of a circle is given by:

Equation: (x - a)2 (y - b)2 r2

Where (a, b) is the center of the circle and r is its radius.

Example Problem: Finding the Center and Radius from a Given Circle Equation

Consider the equation of a circle given as:

x - 32 y - 52 25

To find the center and radius of the circle, we need to compare this equation with the standard form.

Step-by-Step Solution

1. **Identify the forms of the equation:** - The given equation is: x - 32 y - 52 25. Notice that the notation is slightly different from the standard form but still represents a squared term.

2. **Rewrite the equation correctly:** - The correct form of the equation based on the standard form should be: (x - 3)2 (y - 5)2 25.

3. **Compare with the standard equation:** - From the standard equation (x - a)2 (y - b)2 r2, we can see that:

**Identifying the Center and Radius:** - a 3 (The center's x-coordinate)

**b 5 (The center's y-coordinate)

**r2 25 (The radius squared)

4. **Calculate the radius:** - r √25 5 (The radius)

General Approach and Key Points

1. Standard Form: Always start with the standard form of the circle: (x - a)2 (y - b)2 r2.

2. Comparison: Compare the given equation to the standard form to identify a, b, and r2.

3. Finding Radius:** - Once r2 is identified, take the square root to find r (the radius).