Geometry

Question Analysis

The main focus is to determine the center and radius of the circle using the endpoints of its diameter and then write the equation of the circle in standard form.

 

Key Concept Explanation

The standard form of a circle's equation is , where  is the center of the circle and  is the radius. The center of the circle is the midpoint of the diameter, calculated using the midpoint formula  for two points  and . The radius is half the length of the diameter, found using the distance formula .

 

Step - by - Step Solution

1. Find the center of the circle:

Given the endpoints of the diameter  and , use the midpoint formula.

For the  - coordinate of the center, .

For the  - coordinate of the center, . So the center of the circle is .

2. Find the radius of the circle:

Use the distance formula to find the length of the diameter between  and .

.

The radius , and .

3. Write the equation of the circle:

Substitute , , and

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