Problem 43 - College Algebra

Question

Give the center and radius of the circle described by the equation and graph each equation. $$(x-3)^{2}+(y-1)^{2}=36$$

Step-by-Step Solution

Answer

The center of the circle is at (3, 1) and the radius is 6.

Step 1: Identify the center of the circle

The center of the circle can be identified from the terms within the parentheses. Here, \(h\) is the value that subtracts \(x\) and \(k\) is the one that subtracts \(y\). Therefore, the center \((h, k)\) of the circle is \((3, 1)\).

Step 2: Identify the radius of the circle

The radius is given by the square root of the number on the right side of the equation. So here, \(r^2 = 36\), which implies that \(r = \sqrt{36} = 6\). Thus, the radius of the circle is 6.

Step 3: Graph the circle

Plot the center point on the graph at (3, 1). Then, draw a circle with a radius of 6 which expands 6 units left, right, up and down from the center.

Key Concepts

Center of a Circle

Understanding the center of a circle is fundamental in grasping the basics of circle equations. It's the fixed point from which every point on the perimeter of the circle is the same distance away - that distance is known as the radius. In the equation \(x - h)^2 + (y - k)^2 = r^2\), \(h\) and \(k\) represent the x and y coordinates of the center of the circle, respectively.

In our exercise, the equation given was \(x-3)^2+(y-1)^2=36\). Here, the center is found by looking at the numbers that are subtracted from \(x\) and \(y\), which gives us the point \(3, 1\). This indicates that the center of our circle is positioned at 3 units along the x-axis and 1 unit up the y-axis on a coordinate grid. By identifying the center, we establish the reference point from which the circle is drawn and can plot it accurately on a graph.

Radius of a Circle

The radius of a circle is a line segment from the center of the circle to any point on the circle's edge. It plays a crucial role not only in geometry but also in circle equations. In the standard form of the circle equation \(x - h)^2 + (y - k)^2 = r^2\), \(r\) stands for the radius of the circle.

From the given solution of our exercise, \(r^2 = 36\), we find out that the radius (\(r\)) is the square root of the constant term on the equation's right side. Calculating the square root of 36 yields 6, which means our circle has a radius of 6 units. This measurement is vital when you're graphing a circle since it defines the 'size' of your circle and determines how far you'll draw from the center to the circle's boundary.

Graphing Circles

Being able to graph a circle accurately on a coordinate plane is a visual representation of understanding circle equations. Once you know the center and radius, you can sketch the circle by plotting the center and marking points that are the radius's length away in all directions (up, down, left, right and diagonally).

Let's apply this to our exercise where we have already determined the center \(3, 1\) and radius 6. Starting at the center, we'd plot points at the ends of line segments extending 6 units in the cardinal and intermediate directions. Then, you'd connect these points in a smooth, round curve to form the circle. This step-by-step method helps visualize the size and location of the circle on the graph. Remember that every point on the circumference is equidistant from the center, which helps in maintaining the shape of the circle when drawing.