Problem 19 - Intermediate Algebra

Question

Find the center and radius of each circle and graph it. $$ (x+3)^{2}+(y-1)^{2}=16 $$

Step-by-Step Solution

Answer

The circle has a center at \((-3, 1)\) and a radius of 4.

Step 1: Identify Circle Equation

The given equation of the circle is \((x+3)^2 + (y-1)^2 = 16\)which follows the standard form of a circle's equation:\((x-h)^2 + (y-k)^2 = r^2\)where \( (h, k) \) is the center and \( r \) is the radius.

Step 2: Extract Center Coordinates

By comparing the equation \((x+3)^2 + (y-1)^2 = 16\)with the standard form \((x-h)^2 + (y-k)^2 = r^2\),we can identify the center \( (h, k) \) of the circle as \( (-3, 1) \) because:- \(x + 3\) means \(h = -3\).- \(y - 1\) means \(k = 1\).

Step 3: Determine the Radius

In the equation \((x+3)^2 + (y-1)^2 = 16\),the \(r^2\) term equals 16. Therefore, the radius \(r\) is the square root of 16, resulting in:\(r = \sqrt{16} = 4\).

Step 4: Graph the Circle

To graph the circle:1. Plot the center of the circle at the point \((-3, 1)\).2. Using a compass or freehand, draw a circle with a radius of 4 units centered at \((-3, 1)\). The circle will pass through points (1, 1), (-3, 5), (-3, -3), and (-7, 1), which are each 4 units away from the center.

Key Concepts

Standard Form of a Circle

The standard form of a circle's equation is an essential concept to grasp. It is written as \((x - h)^2 + (y - k)^2 = r^2\). This form is very useful because it directly tells us two key details about the circle—its center and its radius.

In this form:

• \( (h, k) \) represents the center of the circle. The \( h \) value affects the horizontal position, while the \( k \) value influences the vertical position.
• \( r \) is the radius of the circle. The squared radius \( r^2 \) is directly given on one side of the equation.

The equation must be in this specific format to easily identify these components. Knowing the standard form allows us to break down any circle equation. Once you see it, you can quickly find out the center and radius of the circle, which is very helpful for graphing or understanding its position on a coordinate plane.

Center of a Circle

Finding the center of a circle from its equation is straightforward once you understand the standard form. The key is to identify the values of \( h \) and \( k \) in the equation \((x - h)^2 + (y - k)^2 = r^2\).

In the given equation from the problem: \((x + 3)^2 + (y - 1)^2 = 16\), we have:

• \( x + 3 \) can be rewritten as \( x - (-3) \), revealing that \( h = -3 \).
• \( y - 1 \) directly shows \( k = 1 \).

Thus, the center of this circle is at the point \((-3, 1)\). Knowing the center location helps us determine the circle's position on a graph, making it easy to plot or draw the circle accurately on paper.

Radius of a Circle

The radius is another critical feature of a circle that can easily be extracted from its equation. In the standard equation, \((x - h)^2 + (y - k)^2 = r^2\), \( r^2 \) represents the radius squared.

In the exercise, the equation \((x + 3)^2 + (y - 1)^2 = 16\) is in standard form. Here, \( r^2 = 16 \).To find \( r \), take the square root of 16:

• \( r = \sqrt{16} = 4 \).

Therefore, the radius of this circle is 4 units. Knowing the radius is essential when sketching or analyzing the circle's size and boundaries. It represents the constant distance from the center point \((-3, 1)\) to any point on the circle's circumference. This distance, or radius, helps in accurately drawing the circle on a plane, ensuring it is perfectly round.