91Ó°ÊÓ

Understanding the equation of a circle is crucial in graphing and identifying its core properties. A circle's equation generally comes in the form: \((x - h)^2 + (y - k)^2 = r^2\). This format is known as the standard form. In this equation, \(h\) and \(k\) represent the coordinates of the center of the circle, and \(r\) is the radius of the circle. Remember that \(r^2\) means the radius squared.
When you have a different format, like \6x^2 + 6y^2 = 216\, your first goal is to convert it to the standard form. This often means simplifying by factoring out common values or manipulating terms.

Identifying the center and radius of a circle is key to graphing it accurately. After converting your circle's equation by simplifying or rearranging, you can compare it with the standard form to find the center and radius.
For example, in the equation \6x^2 + 6y^2 = 216\, simplifying it by dividing by 6 gives us: \(x^2 + y^2 = 36\). This can be rewritten as \((x - 0)^2 + (y - 0)^2 = 6^2\), meaning that the center \(h, k\) is at (0, 0) and the radius \(r\) is 6.
So, always follow these steps:

• Simplify or rearrange the given equation.
• Identify \(h, k\), the center.
• Determine \(r\), the radius, by taking the square root of the value on the right side of the equation.

The standard form of a circle provides an easy way to identify the circle's center and radius directly. The standard equation of a circle is: \((x - h)^2 + (y - k)^2 = r^2\), where:

• \(h, k\) is the center.
• \r\ is the radius.

This form makes it straightforward to graph the circle since you immediately know where the center is and how far out the circle extends.
To transform an expanded circle equation to this form, like we did with \6x^2 + 6y^2 = 216\, divide by constants to isolate the squared terms and match the standard structure. After simplifying, you can directly read off the center and radius from the equation. For example, \x^2 + y^2 = 36\ becomes \((x-0)^2 + (y-0)^2 = 6^2\), clearly showing a center at (0, 0) and a radius of 6.