Geometry

Question Analysis

The main focus is to convert the given equation of the circle from the general form  to the standard form .

 

Key Concept Explanation

The general form of a circle's equation is . To obtain the standard form, we use the method of completing the square. For the  terms , we take half of , square it and add it to both sides of the equation. Similarly, for the  terms , we take half of , square it and add it to both sides. The standard form  represents a circle with center  and radius .

 

Step-by-Step Solution

1. Given the equation .

2. Complete the square for the  terms:

The coefficient of  is . Half of  is , and . Add  to both sides of the equation.

Complete the square for the  terms: The coefficient of  is . Half of  is , and . Add  to both sides of the equation.

The equation becomes .

3. Rewrite the equation:

Factor the  and  terms. We have .

 

Option Analysis

A.

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