Problem 71 - Algebra 2

Question

What is the radius of the circle with equation \((x+5)^{2}+(y-3)^{2}=144 ?\)

Step-by-Step Solution

Answer

The radius of the circle is 12.

Step 1: Identify the form of the equation

Rewrite the given equation in the standard form of a circle's equation. Our given equation is \((x+5)^{2}+(y-3)^{2}=144 .\) The general equation of a circle 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 with the standard form

By comparing the standard form with our given equation, we find the following correspondences: The center (h, k) corresponds to (-5, 3) and the r^2 corresponds to 144.

Step 3: Calculate the radius

Knowing that the value of r squared is 144, we compute the square root to get the radius r. Thus, the radius r of the circle is \( \sqrt{144} = 12 \).

Key Concepts

Radius of a Circle

The radius of a circle is a crucial measurement that defines the size of the circle. It is the distance from any point on the circle's edge to its center. In mathematical terms, if a circle is centered at point \( (h, k) \), and a point \( (x, y) \) lies on the circle, the distance between \( (x, y) \) and \( (h, k) \) is the radius, \( r \).
In the context of circle equations, the radius is identified in the equation written in the form \( (x-h)^2 + (y-k)^2 = r^2 \).
For example, in the given problem, we have an equation \( (x+5)^2 + (y-3)^2 = 144 \). After identifying that \( r^2 = 144 \), we calculate the radius by taking the square root, giving us \( r = 12 \).

• Radius gives us insight into the circle's size.
• It is half the length of the diameter, the longest distance across the circle.
• Understanding the radius helps in calculating circle-related metrics like circumference and area.

Standard Form of a Circle

The standard form of a circle helps us express the equation of a circle in a way that makes it easy to identify important properties like the center and the radius.
The standard form of a circle's equation is: \( (x-h)^2 + (y-k)^2 = r^2 \).
In this equation:

• \( (h, k) \) are the coordinates of the circle's center.
• \( r \) represents the radius of the circle.

We can rewrite any circle equation into this standard form to easily recognize these features. For instance, the equation given in the practice, \( (x+5)^2 + (y-3)^2 = 144 \), is already in the standard form, helping us immediately identify:

• The center of the circle: \( (-5, 3) \).
• The radius squared: \( r^2 = 144 \), leading to a radius of \( r = 12 \).

Understanding this form is essential for solving problems related to circles efficiently.

Center of a Circle

The center of a circle is the point equidistant from all points on the boundary of the circle. In the standard form of a circle's equation \( (x-h)^2 + (y-k)^2 = r^2 \), the center is represented by the coordinates \( (h, k) \).
In simple terms, it is the 'middle' point of the circle.
Recognizing the center is vital as it serves as the point of rotation, reference, or symmetry for the entire circle.
Given the equation \( (x+5)^2 + (y-3)^2 = 144 \), we can easily identify the center by looking at:

• The coefficients inside the squared terms \( (x+5) \) and \( (y-3) \).
• The center calculated from this equation is \( (-5, 3) \), which is obtained by applying \( (x-h) \) and \( (y-k) \). Remember to change the sign when determining the center.

Knowing the center allows us to fully utilize the radius for things like plotting the circle or determining its position relative to other geometric figures.