Problem 44 - College Algebra

Question

Give the center and radius of the circle described by the equation and graph each equation. $$ (x-2)^{2}+(y-3)^{2}=16 $$

Step-by-Step Solution

Answer

The center of the circle is (2,3) and the radius is 4.

Step 1: Identify the circle's center

The center (h,k) can be identified by looking at the terms in the parentheses in the given equation. In the equation \((x-2)^2 + (y-3)^2 = 16\), the center (h,k) is (2,3).

Step 2: Identify the circle's radius

The radius can be identified by taking the square root of the number on the right-hand side of the equation. In this case, the square root of 16 is 4. So, the radius of the circle is 4.

Step 3: Graph the circle

To graph the circle, locate the center point (2,3) on an x-y coordinate grid. Then, draw a circle with a radius of 4 units centered at that point. Be sure to use a consistent scale so that the circle appears round rather than elliptical.

Key Concepts

Understanding the Circle's Center

The center of a circle is a crucial point that helps define its location on a coordinate plane. To find it for a given circle's equation, observe the structure of the equation, specifically the expression within the parentheses.

For example, consider the equation \[(x-2)^2 + (y-3)^2 = 16.\]The center of this circle, denoted as \((h, k)\), is found by analyzing the values subtracted from each variable within the parentheses. Here, \(h = 2\) and \(k = 3\), making the center \((2, 3)\).

• The expression \((x - h)^2 + (y - k)^2\) helps locate the circle's center \((h, k)\).
• The signs inside the parentheses are reversed. So, if the equation is \((x - 2)^2\), the \(x\)-coordinate of the center is 2.
• Always match the values from inside the parentheses to obtain the circle's center.

This systematic approach ensures that pinpointing the center is straightforward and error-free.

Calculating the Circle's Radius

The radius of a circle represents the distance from its center to any point on its circumference. In mathematical terms, it is half the diameter of the circle.

When given an equation of a circle, like \((x-2)^2 + (y-3)^2 = 16\), identifying the radius involves focusing on the number on the equation's right-hand side. This number represents the radius squared.

• To find the radius, you take the square root of this number.
• In our example, the right side is 16. Thus, \(\sqrt{16} = 4\).
• This calculation shows that the circle's radius is 4 units.

Being able to derive the radius can greatly assist when graphing or analyzing the circle's size and characteristics. It's important to remember that this distance is consistent from the center to any boundary point on the circle.

Graphing the Circle

Successfully graphing a circle involves a few basic steps, which begin with understanding the circle's equation. Once you have determined the circle's center and radius, you can accurately plot it on a coordinate grid.

• Start by marking the center point on the graph. For our example, the center \((2, 3)\) is plotted by moving 2 units along the x-axis and 3 units along the y-axis.
• From this center, measure out the radius, which is 4 units in this case, in all directions to establish the circumference.
• Using a compass or freehand, draw a circle around the center point that touches each radius endpoint.

The key to a proper graph is maintaining consistent scale so the circle appears correctly round, not stretched into an ellipse. By following these simple steps, you will accurately represent the circle described by its equation.