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Chapter 13: Problem 18

Find the center and radius of each circle. Then graph the circle. $$ (x-3)^{2}+(y-2)^{2}=4 $$

Short Answer

Expert verified
Center: (3, 2), Radius: 2

Step by step solution

01

Identify the standard form

The given equation \( (x - 3)^{2} + (y - 2)^{2} = 4 \) is already in the standard form of a circle equation \[ (x - h)^{2} + (y - k)^{2} = r^{2} \]
02

Determine the center

Compare the given equation with the standard form and identify \( h \) and \( k \). Here, \( h = 3 \) and \( k = 2 \). Therefore, the center of the circle is \( (3, 2) \).
03

Determine the radius

Identify \( r^{2} \) from the equation. In this case, \( r^{2} = 4 \). Taking the square root of both sides, \( r = 2 \). So, the radius of the circle is \( 2 \).
04

Graph the circle

To graph the circle, plot the center at \( (3, 2) \) on the coordinate plane. Then from the center, measure a distance of \( 2 \) units upwards, downwards, leftwards, and rightwards to plot points on the circumference. Finally, draw the circle passing through these points.

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Key Concepts

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

standard form
The standard form of a circle's equation is essential for understanding and performing various tasks related to circles in mathematics. The standard form is expressed as \((x - h)^{2} + (y - k)^{2} = r^{2}\). Here, \((h, k)\) represents the center of the circle, and \(r\) is the radius. This form allows us to easily identify the circle's essential characteristics directly from the equation. From here, we can determine the center and the radius with minimal calculations.
center of a circle
The center of a circle is a vital concept to understand. It is represented in the equation \((x - h)^{2} + (y - k)^{2} = r^{2}\) by the coordinates \((h, k)\). In the example given \((x - 3)^{2} + (y - 2)^{2} = 4\), we can identify the center by comparing it to the standard form. Here, \(h = 3\) and \(k = 2\), so the center is \((3, 2)\). The coordinates give us a reference point from which the circle extends uniformly in all directions.
radius of a circle
The radius of a circle is the distance from the center to any point on its circumference. This is indicated by the term \(r^{2}\) in the standard form \((x - h)^{2} + (y - k)^{2} = r^{2}\). To find the radius in the equation \((x - 3)^{2} + (y - 2)^{2} = 4\), we recognize \(r^{2} = 4\). By taking the square root of both sides, we get \(r = 2\). Hence, the radius of the circle is \2 units\. Knowing the radius allows us to draw the circle accurately.
graphing circles
Graphing a circle involves plotting the center and using the radius to define the circle's boundary. For the equation \((x - 3)^{2} + (y - 2)^{2} = 4\), the center is \(3, 2\) and the radius is \2\. On a coordinate plane, follow these steps:
  • Plot the center at \( (3, 2)\).
  • Measure \2\ units from the center upwards, downwards, leftwards, and rightwards to plot key boundary points.
  • Connect these points to form a smooth circle.
With these steps, you can visualize how the circle extends evenly from the center.

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