91Ó°ÊÓ

A circle equation describes the set of all points that are equidistant from a fixed point called the center. In the given original exercise, we started with an equation involving both \(x^2\) and \(y^2\). Such equations generally describe conic sections, which include circles, ellipses, parabolas, and hyperbolas. The general form of a circle's equation is often written as \(x^2 + y^2 + Cx + Dy + E = 0\), where \(C\), \(D\), and \(E\) are constants. However, to easily identify the properties of the circle, the equation should be rewritten in a form that allows identification of the center and radius. The process of completing the square, as executed in the steps of the exercise, helps convert the equation into its standard form. This involves rearranging and manipulating the equation so it becomes evident where the center and radius are located.

The center and radius of a circle are fundamental in understanding and graphing the circle.

• The center of a circle is denoted generally as \((a, b)\), and it marks the point around which the circle is perfectly symmetrical.
• The radius is the constant distance between the center and any point on the circle's boundary.

In our exercise, by rewriting the original equation, we were able to glean the circle's center and its radius. After completing the square, the equation was transformed into the form \((x - a)^2 + (y - b)^2 = r^2\). For our problem, this form highlighted that the center of the circle is \((0, 3)\) and the radius is 4, as shown by the term \(16\), which is equal to \(r^2\). Thus, \(r = \sqrt{16} = 4\). These values are crucial as they dictate the circle's position and size when graphed.

The standard form of a circle is crucial for easily identifying the geometrical properties of a circle in algebraic form. The standard form is expressed generally as \((x - a)^2 + (y - b)^2 = r^2\), where:

• \((a, b)\) is the center of the circle.
• \(r\) is the radius.

To transform a given quadratic equation into this standard form, we often use a technique called completing the square. This technique reorganizes terms and involves adding specific values to both sides of the equation to make perfect squares from binomials. For the original problem, completing the square turned our initial equation into the standard form, revealing its center and radius. This transformation allows us to clearly see the properties of the circle, facilitating not only understanding but also helping in drawing the graphical representation of the circle.