Problem 16 - Intermediate Algebra

Question

Find the center and radius of each circle and graph it. $$ x^{2}+y^{2}=16 $$

Step-by-Step Solution

Answer

The center of the circle is \((0, 0)\) and the radius is 4.

Step 1: Identify the Standard Form of a Circle

The equation of a circle in its standard form is \((x-a)^2 + (y-b)^2 = r^2\), where \((a, b)\) is the center and \(r\) is the radius.

Step 2: Compare and Identify Parameters

Compare the given equation \(x^2 + y^2 = 16\) with the standard form \((x-a)^2 + (y-b)^2 = r^2\). You can see that \(a = 0\), \(b = 0\), and \(r^2 = 16\).

Step 3: Calculate the Radius

Since \(r^2 = 16\), take the square root of both sides to find \(r\). So, \(r = \sqrt{16} = 4\).

Step 4: Determine the Center of the Circle

From Step 2, we identified that \(a = 0\) and \(b = 0\), which means that the center of the circle is at the origin \((0, 0)\).

Step 5: Graph the Circle

To graph the circle, plot the center at \((0, 0)\) on a coordinate plane. Then, using the radius \(r = 4\), draw a circle that is equally distant (4 units) from the center in all directions.

Key Concepts

Standard Form of a Circle

The standard form of a circle's equation is fundamental when working with circles in algebra. It is a structured way to express a circle, which makes it clear where the circle is positioned and how big it is. The equation is given by:

• \((x-a)^2 + (y-b)^2 = r^2\)

In this equation:

• \((a, b)\) represents the coordinates of the circle's center.
• \(r\) represents the circle's radius.

Understanding this form allows you to easily identify the center and radius of a circle and to graph it accurately. It is designed to precisely capture the simple geometric properties of a circle on a coordinate plane.

Center of a Circle

The center of a circle is a critical point that helps define the circle's size and location on the coordinate plane. Using the standard form of the circle,

• \((x-a)^2 + (y-b)^2 = r^2\)

The center is represented by the coordinates \((a, b)\). In our exercise, the equation is \(x^2 + y^2 = 16\). Here, it becomes apparent when compared to the standard form that:

• \(a = 0\)
• \(b = 0\)

Thus, the center of the circle is at the origin, \((0, 0)\). This central point is where all radii of the circle extend out to the circumference, confirming it as the symmetrical focal point of the circle.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its circumference. This distance remains constant around the entire circle. In the standard equation,

• \((x-a)^2 + (y-b)^2 = r^2\)

We calculate the radius \(r\) by taking the square root of \(r^2\). For the exercise given, the equation is \(x^2 + y^2 = 16\). Here, you can identify:

• \(r^2 = 16\)
• Therefore, \(r = \sqrt{16} = 4\)

This means the radius is 4 units. The radius is vital as it not only defines the size of the circle but also assists in plotting the circle accurately during graphing.

Graphing Circles

Graphing a circle involves plotting its center and drawing points equidistant from this center, forming a circular shape. Once the center and radius are known, graphing becomes simple:

• First, plot the center of the circle on the coordinate plane. For our example, this is the origin, \((0, 0)\).
• Next, using the radius \(r = 4\), mark points that are 4 units away from the center in all directions.
• Connect these points smoothly to form a circle.

Graphing visually represents the mathematical relationship expressed in the equation and allows for an intuitive understanding of how changes to the equation affect the circle's placement and size.It's important to understand that every point on the circle is exactly 4 units away from the center, creating a perfectly round shape. This helps reinforce the relationships defined in the standard form and exhibits the symmetrical nature of circles.