Understanding the Circle's Equation and Its Components

The equation of a circle is a fundamental concept in mathematics, particularly in geometry. Given an equation such as x2 y2 - 2x - 4y 1 0, one of the primary tasks is to determine the center and the radius of the circle. This article explores various methods to achieve this, emphasizing the importance of the concept of completing the square.

Method 1: General Equation of a Circle

The general equation of a circle is x2 y2 - 2ax - 2by a2 b2 - R2 0, where (a, b) represents the coordinates of the center, and R is the radius.

Given x2 y2 - 2x - 4y 1 0, we can rewrite it in the form of the general equation:

x2 y2 - 2*1*x - 2*2*y 12 22 - R2 0

Matching the terms, we have a 1 and b 2.

R2 12 22 - 1 4 - 1 3

Therefore, R u221A3, and the center is (1, 2).

Method 2: Completing the Square

The process of completing the square is often used to transform a general polynomial equation into a more recognizable form. For the equation x2 y2 - 2x - 4y 1 0, we can complete the square for both x and y terms.

x2 - 2x 1 y2 - 4y 4 3

(x - 1)2 (y - 2)2 22

This shows that the circle is centered at (1, 2) and has a radius of 2 units.

Method 3: Simplifying the Equation Directly

Another approach is to simplify the given equation directly by completing the square for each term.

x2 - 2x 1 - 1 y2 - 4y 4 - 4 1 0

(x - 1)2 (y - 2)2 4

This confirms that the center of the circle is at the point (1, 2) and the radius is 2 units.

Conclusion and Adapting to General Form

The general form of a circle's equation can be adapted to the standard form using the formula:

Center ( -g, -f )

Radius u221A(g2 f2 - c)

For the given equation, we have:

2g -2 rarr; g -1

2f -4 rarr; f -2

c 1

Substituting these values into the formulas, we get:

Center (1, 2)

Radius u221A((-1)2 (-2)2 - 1) u221A(1 4 - 1) u221A4 2

Thus, the center and radius of the circle are clearly identified as (1, 2) and 2 units, respectively.

Understanding how to solve these circle equations by completing the square is a crucial skill in advanced mathematics. This approach not only simplifies the problem but also provides a deeper insight into the geometric properties of circles.