Problem 78 - College Algebra Essentials

Question

Give an example of a circle's equation in standard form Describe how to find the center and radius for this circle.

Step-by-Step Solution

Answer

The center of the circle is at (3, 4) and the radius is 7 units.

Step 1: Provide an example of a circle's equation in standard form

An example of a circle's equation in standard form can be \( (x- 3)^{2} + (y - 4)^{2} = 49 \).

Step 2: Identify the center of the circle

To identify the center of the circle for this equation, check the values of h and k, the pair inside the brackets along with x and y in the equation above. Therefore, the center of the circle is (3, 4).

Step 3: Identify the radius of the circle

The radius of the circle is obtained from the value after the equal sign. That is, r². Here, r² is 49. To get the radius (r), we need to take the square root of 49, r = \( \sqrt{49} \), thus r = 7 is the radius of the circle.

Key Concepts

Standard Form

A circle's equation in standard form is structured as follows: \((x-h)^2 + (y-k)^2 = r^2\). In this equation:

• \(h \) and \(k\) are constants that represent the coordinates of the circle's center.
• \(r\) is the radius of the circle.

The standard form equation of a circle allows us to easily identify both the center and the radius, as these elements are clearly indicated in the equation.
For example, in the equation \((x-3)^2 + (y-4)^2 = 49\), the numbers inside the parentheses with \(x\) and \(y\) correspond to \(h\) and \(k\), while the number 49 on the right side represents the square of the radius \(r^2\). To fully utilize the standard form, simply examine the equation components: this will help uncover key properties of the circle quickly.

Center of the Circle

The center of a circle in its standard form equation is found by examining the constants \(h\) and \(k\) in the expression \((x-h)^2 + (y-k)^2 = r^2\). The center is represented by the point \((h, k)\).
To locate the center, adjust the signs from the parentheses in the standard form equation. For instance, look at the example equation \((x-3)^2 + (y-4)^2 = 49\). The values inside the parentheses are \(x-3\) and \(y-4\):

• The number 3 with an opposite sign becomes \(3\),
• The number 4 with an opposite sign becomes \(4\).

This shows that the center of the circle is at the coordinate point \((3, 4)\).
It's crucial to translate the signs accurately, as misinterpretation here may lead to errors in identifying the position of the circle within the coordinate plane.

Radius of the Circle

The radius is another fundamental aspect of the circle’s equation in standard form. It's derived from the number on the right side of the equation, representing \(r^2\). To find the radius, extract the square root of this number.
In our example, \((x-3)^2 + (y-4)^2 = 49\), the number 49 is equivalent to \(r^2\). Therefore, you take the square root of 49 to find \(r\):

• \(\sqrt{49} = 7\)

Thus, the radius \(r\) is 7.
Understanding this process is important for determining the size of the circle. Since the radius affects the circle's size dramatically, make sure to accurately calculate it from the \(r^2\) value. Knowing the radius helps in visualizing and plotting the circle on a graph, making calculations regarding the circle very precise.