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Chapter 14: Problem 5

Identify the center and radius of each circle and graph. $$(x-5)^{2}+(y-3)^{2}=1$$

Short Answer

Expert verified
The center of the circle is at point (5, 3), and the radius is 1. To graph the circle, plot the center at (5, 3) and draw a circle with a radius of 1 unit around the center point.

Step by step solution

01

In this problem, the given equation is \((x-5)^{2}+(y-3)^{2}=1\). We can observe that this equation is in the form \((x-a)^{2}+(y-b)^{2}=r^{2}\). #Step 2: Identify the coordinates of the center (a, b)#

We can see that in the given equation, the values of \(a\) and \(b\) are 5 and 3, respectively. Therefore, the center of the circle is at point \((a, b) = (5, 3)\). #Step 3: Identify the radius r#
02

The given equation is in the form \((x-a)^{2}+(y-b)^{2}=r^{2}\), and we can observe that \(r^{2}=1\). To find the radius, we need to find the square root of \(r^{2}\): \(r=\sqrt{1}\). Therefore, the radius of the circle is \(r=1\). #Step 4: Graph the circle#

To graph the circle, follow these steps: 1. Plot the center of the circle at point (5, 3) on the coordinate plane. 2. From the center, measure a distance of 1 unit (the radius) in all directions and draw a circle through those points. The resulting graph will represent the circle with center at (5, 3) and a radius of 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Center of a circle
To determine the center of a circle from its equation, we use the standard form of the circle's equation: \(x-a\)^2 + \(y-b\)^2 = r^2. This equation tells us that the center is at point \(a, b\) on the coordinate plane. The values of \(a\) and \(b\) are derived directly from the transformation of the equation's x and y terms: \(x-a\) and \(y-b\).

  • In the equation \((x-5)^{2}+(y-3)^{2}=1\), the center is at (5, 3).
  • The signs before 5 and 3 indicate shifts from the origin in the positive x and positive y directions, respectively.

Understanding the concept of the center helps in visualizing where the circle is "anchored" on a plane, acting as the heart of the circle.
Radius of a circle
The radius is the distance from the center of the circle to any point on the circle itself. It is represented as \(r\) in the circle's equation.

To find the radius from the given circle equation, \(x-a\)^2 + \(y-b\)^2 = r^2, take note of the value of \(r^2\) on the right side of the equation.

  • For instance, in the equation \((x-5)^{2}+(y-3)^{2}=1\), we have \(r^2 = 1\).
  • By taking the square root of 1, we find that \(r = \sqrt{1} = 1\).

The radius determines the size of the circle. A bigger radius means a larger circle, whereas a smaller radius means a more compact circle.
Graphing circles
Graphing a circle involves plotting the center and using the radius to draw the circle's boundary on the coordinate plane. This process takes the algebraic information from the equation and translates it into a visual representation.

Here鈥檚 how to do it:
  • First, identify and plot the center of the circle. For our example, this is the point (5, 3).
  • Next, use the radius to measure out from the center in all directions. In this case, you move 1 unit left, right, up, and down from the center.
  • Finally, draw a smooth, continuous curve through these points to form the circle.
Graphing circles helps in developing spatial understanding and makes abstract mathematical concepts more tangible. Working with graphs is a great way to verify the correctness of your calculations about the circle's center and radius.

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