Problem 62 - College Algebra with Modeling and Visualization

Question

Find the center and radius of the circle. $$ x^{2}+y^{2}=20 $$

Step-by-Step Solution

Answer

The center is (0, 0) and the radius is \(2\sqrt{5}\).

Step 1: Identify the standard form of the circle equation

The general standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) are the coordinates of the center of the circle and \(r\) is the radius.

Step 2: Compare the given equation to the standard form

The given equation is \(x^2 + y^2 = 20\). Notice that it matches \((x-0)^2 + (y-0)^2 = 20\), indicating \(h = 0\), \(k = 0\), and \(r^2 = 20\).

Step 3: Determine the center of the circle

Since the equation \((x-0)^2 + (y-0)^2 = 20\) results in \(h = 0\) and \(k = 0\), the center of the circle is \((0, 0)\).

Step 4: Calculate the radius

To find the radius, take the square root of \(r^2\). Thus, \(r = \sqrt{20}\). Simplified, the radius is \(r = 2\sqrt{5}\).

Key Concepts

Standard Form of a Circle

The standard form of a circle's equation is a formula used to describe a circle in the coordinate plane. It is written as \((x-h)^2 + (y-k)^2 = r^2\). In this form,

• \((h, k)\) represents the center of the circle, and
• \(r\) is the radius of the circle.

This equation is particularly useful because it allows us to easily identify the center and radius of a circle by simply looking at the equation. For example, consider the equation \(x^2 + y^2 = 20\). By rewriting it, we can see it matches the standard form with \((h, k) = (0, 0)\) and \(r^2 = 20\). This means the circle is centered at the origin and gives us information needed for further calculations.

Center of a Circle

Identifying the center of a circle is crucial when working with circle equations. The center

• is represented by the point \((h, k)\) in the standard form \((x-h)^2 + (y-k)^2 = r^2\).
• This point is the location from which every point on the circle is equidistant.

In our specific example, \(x^2 + y^2 = 20\), you can see there are no values subtracted from \(x\) or \(y\). This implies \(h = 0\) and \(k = 0\), identifying the center as the origin \((0, 0)\). Knowing the center is important because it allows us to determine where the circle is positioned on the coordinate plane.

Radius Calculation

The radius of a circle is the distance from the center to any point on the circle. It is denoted by \(r\) in the standard form \((x-h)^2 + (y-k)^2 = r^2\). In the equation we have, \(x^2 + y^2 = 20\), the term \(20\) is actually \(r^2\). To find the radius, we take the square root of this term. Thus, \(r = \sqrt{20}\), which can be simplified to \(r = 2\sqrt{5}\). This means the radius is approximately 4.47 units. Calculating the radius helps us understand the size of the circle and how far it extends from its center.