Problem 20 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the equation of the circle in standard form. Then identify its center and radius. $$\frac{1}{9} x^{2}+\frac{1}{9} y^{2}=1$$

Step-by-Step Solution

Answer

The center of the circle is at the origin (0,0), and the radius is 3.

Step 1: Identify the Center

The center of the circle in standard form \((x-h)^{2}+(y-k)^{2}=r^{2}\) is at the point (h,k). Here, the equation of the circle doesn't contain any x or y terms, meaning the center of the circle is at the origin, (0,0).

Step 2: Identify the Radius

The radius of the circle in standard form is determined by r in equation \((x-h)^{2}+(y-k)^{2}=r^{2}\). Here, the equation of the circle is \(\frac{1}{9} x^{2}+\frac{1}{9} y^{2}=1\). To get the radius in the form of \(r^{2}\), we can rewrite it as \(x^{2}+y^{2}=9\). The radius r is the square root of this term, which we find to be r = 3.

Key Concepts

Circle Equations

Circle equations are essential for understanding the properties of circles and solving problems related to them. To start, each circle can be represented by an equation that showcases its symmetry. One common form of a circle's equation is the standard form, which appears as \( (x-h)^2 + (y-k)^2 = r^2 \).

In this formula, \(x \) and \(y \) are the variables representing the coordinates of any point on the circle, while \(h \) and \(k \) denote the coordinates of the circle's center. The term \(r\) stands for the radius of the circle, which is the constant distance from the center to any point on the circle's circumference.

When analyzing the given exercise, \(\frac{1}{9} x^{2}+\frac{1}{9} y^{2}=1\), we notice this equation is nearly in standard form but with a coefficient of \(\frac{1}{9}\) on the squared terms. By multiplying both sides by 9, we bring it to the standard form \(x^{2}+y^{2}=9\), immediately making it easier to identify the circle's properties.

Center of a Circle

The center of a circle is the fixed point from which every point on the circumference is equidistant. In the standard form equation of a circle \( (x-h)^2 + (y-k)^2 = r^2 \), the center is represented by the point \( (h, k) \).

For our exercise, since the equation lacks any \(x \) or \(y \) terms outside the squared components, it suggests that the center of the circle is at the origin, which is \( (0, 0) \). This is common for circles that are centered at the origin of a coordinate plane, where \(h \) and \(k \) both equal zero, simplifying the equation to \( x^2 + y^2 = r^2 \), where the center \( (h, k) \) is implied to be \( (0, 0) \).

Understanding the center's location is crucial for graphing the circle and solving various geometric problems, such as finding the circle's intersection with other shapes or lines.

Radius of a Circle

The radius of a circle is one of its defining features. It is the distance from the center of the circle to any point on the circumference. In the equation \( (x-h)^2 + (y-k)^2 = r^2 \), \(r\) stands for the radius, and its value is always positive.

For the exercise provided, after rearranging the original equation to its standard form, we have \(x^{2}+y^{2}=9\). In this equation, \(r^2 \) is represented by the number 9. To determine the radius, we take the square root of 9, which gives us \( r = 3 \).

The radius is not just a measurement; it's used in multiple area and circumference calculations and is essential for understanding the scale and size of the circle. Additionally, the radius can help determine the circle's relation to other shapes in coordinate geometry and in real-world contexts, such as wheel size, circular tracks, and more.