Problem 78 - Precalculus Essentials

Question

Give an example of a circle's equation in standard form. Describe how to find the center and radius for this circle.

Step-by-Step Solution

Answer

For the given example circle equation \( (x-3)² + (y-5)² = 16 \), the center of the circle is at point (3, 5) and the radius is 4.

Step 1: Circle's Equation in Standard Form

The general form for the equation of a circle is \( (x-a)² + (y-b)² = r² \), where (a,b) represents the center and r represents the radius of the circle. Let's take an example circle equation, such as \( (x-3)² + (y-5)² = 16 \).

Step 2: Identifying the Center of the Circle

The center of the circle is given by the coordinates (a, b). From the equation \( (x-3)² + (y-5)² = 16 \), the center can be extracted to be at the point (3, 5).

Step 3: Identifying the Radius of the Circle

The radius of the circle can be found by taking the square root of the right side of the equation. From the equation \( (x-3)² + (y-5)² = 16 \), the radius is \( \sqrt{16} = 4 \).

Step 4: Summary

So, for the example circle equation \( (x-3)² + (y-5)² = 16 \), the center of the circle is at point (3, 5) and the radius is 4.