91Ó°ÊÓ

The equation of a circle in its standard form is given by \[ x^2 + y^2 = r^2 \]where \(r\) represents the radius of the circle. In this equation, the center of the circle is located at the origin

• (0,0)

This formula is derived from the Pythagorean theorem, assuming the circle's center is the origin and any point

• (x,y)

on the circle satisfies this equation.
To identify a circle from an equation, you need to first place it in the standard form. For example, in the original equation, \(9x^2 + 9y^2 = 4\), we can see every term multiplied by 9.
By dividing the whole equation by 9, it transforms into \( x^2 + y^2 = \frac{4}{9}\), which nicely fits the standard form and confirms it represents a circle.

The radius of a circle is the distance from the center to any point on its circumference. To find the radius in a given circle equation:

• First, convert the equation to the standard form \(x^2 + y^2 = r^2\).
• The term on the right side of the equation, \(r^2\), is used to calculate the radius \(r\).

In our example, the simplified equation is \(x^2 + y^2 = \frac{4}{9}\). Hence, \(r^2 = \frac{4}{9}\).
This gives us the radius:\[ r = \sqrt{\frac{4}{9}} = \frac{2}{3} \]
Therefore, the radius is \(\frac{2}{3}\) units. Calculating the square root is essential to understand how much the circle extends from its center. This process is critical for accurately sketching the graph.

Graphing a circle involves plotting it on a coordinate plane based on its equation in the standard form. Here’s how you can do it step by step:

• Identify the center of the circle. For the equation \(x^2 + y^2 = \frac{4}{9}\), the circle's center is at (0,0).
• Identify the radius, which we calculated as \(\frac{2}{3}\).
• Properly draw the coordinate axis. The origin (0,0) should be marked clearly at the center.
• Using the radius, mark points at this distance around the center. You'll reach points such as (\(\frac{2}{3}, 0\)), (\(0, \frac{2}{3}\)) on the coordinate plane.

Once these points are marked, you can sketch a circle passing through them centered at the origin.
This visual representation validates that every point \( (x, y) \) on the circle satisfies the equation \(x^2 + y^2 = \frac{4}{9}\). Coordinate graphing this way helps visualize the actual geometry of the circle and links algebraic equations to geometric figures.