Problem 22 - Algebra 2

Question

Find the center and radius of the circle with the given equation. Then graph the circle. $$ (x-3)^{2}+(y-1)^{2}=25 $$

Step-by-Step Solution

Answer

The center is (3, 1) and the radius is 5.

Step 1: Identify the Standard Form of a Circle Equation

The given equation is \((x-3)^2 + (y-1)^2 = 25\). The standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Determine the Center of the Circle

Compare the given equation \((x-3)^2 + (y-1)^2 = 25\) with the standard form \((x-h)^2 + (y-k)^2 = r^2\). We see that \(h = 3\) and \(k = 1\). Thus, the center of the circle is \((3, 1)\).

Step 3: Determine the Radius of the Circle

According to the equation \((x-3)^2 + (y-1)^2 = 25\), \(r^2 = 25\). To find the radius \(r\), take the square root: \(r = \sqrt{25} = 5\). The radius is 5 units.

Step 4: Graph the Circle

To graph the circle, plot the center at \((3, 1)\) on a coordinate plane. From this center, draw a circle with a radius of 5 units. Make sure the circle is equidistant from the center in all directions, creating a symmetric shape.

Key Concepts

Center of a Circle

In the equation of a circle given by \[(x-h)^2 + (y-k)^2 = r^2,\]the point \((h, k)\) represents the center of the circle. Understanding the center of a circle is crucial because it defines the origin from which all points on the circle are equidistant. In this exercise, we were given the equation: \[(x-3)^2 + (y-1)^2 = 25.\]To find the center, we compare it with the standard form \[(x-h)^2 + (y-k)^2.\]

• The term \((x-3)^2\) tells us that \(h = 3\).
• The term \((y-1)^2\) indicates \(k = 1\).

So, the center of this circle is located at \((3, 1)\). This information is essential for graphing and understanding the circle's position on a coordinate plane.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. In the circle's equation \[(x-h)^2 + (y-k)^2 = r^2,\]\(r\) represents the radius. From the problem:\[(x-3)^2 + (y-1)^2 = 25,\]we notice that \\[r^2 = 25.\]To find the radius \(r\), we need to take the square root of 25: \\[r = \sqrt{25} = 5.\]The radius is 5 units. Knowing the radius is vital because it dictates how large the circle is when you graph it.

Graphing Circles

Graphing a circle involves placing it correctly on a coordinate plane. Using both the center and the radius, we can accurately depict the circle.

• Firstly, plot the center of the circle. For this problem, the center is at \((3, 1)\).
• Next, use the radius to determine the size of the circle. With a radius of 5, you will move 5 units up, down, left, and right from the center to mark the circle's boundary points.

Ensure your plotting shows that all these boundary points are the same distance from the center. The result should be a perfectly round shape, illustrating the concept that every point on the circle is equidistant from the center, confirming our findings from the equation.