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Chapter 7: Problem 149

Graph each circle. Identify the center and the radius. \((x-2)^{2}+(y-3)^{2}=4\)

Short Answer

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

Step by step solution

01

- Identify the general form of the circle equation

The general form of the equation of a circle is \( (x-h)^{2}+(y-k)^{2}=r^{2} \), where \( (h,k) \) is the center and \( r \) is the radius.
02

- Extract the center coordinates

Compare the given equation \( (x-2)^{2}+(y-3)^{2}=4 \) with the general form. Here, \( h=2 \) and \( k=3 \), so the center of the circle is (2, 3).
03

- Find the radius

The radius is \( r \) which is the square root of the right-hand side of the equation. Here, \( r^{2}=4 \) so \( r= \sqrt{4} \), which simplifies to \( r=2 \).
04

- Graph the circle

Draw a coordinate plane. Plot the center at (2, 3). From this center, draw a circle with a radius of 2 units. The circle should extend 2 units in all directions (up, down, left, and right) from the center.

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Key Concepts

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

Circle Equation
The equation of a circle is often written in a specific format which helps us easily identify its key features. This format is known as the standard form of a circle equation. The standard form is \[ (x-h)^{2}+(y-k)^{2}=r^{2} \]Here,
  • \(h\) and \(k\) are the coordinates of the center of the circle.
  • \(r\) is the radius of the circle.
Recognizing this form helps in sketching and understanding the circle. Any circle equation can be identified and graphed if we rewrite it in this standard form. Simply compare each part of your given circle equation to break down its components.
Center of Circle
The center of a circle is a key feature and is given by the coordinates \( (h, k) \) in the standard form equation. For the given problem, we have \[ (x-2)^{2}+(y-3)^{2}=4 \]By matching this with \[ (x-h)^{2}+(y-k)^{2}=r^{2} \]we see that \(h=2\) and \(k=3\). Thus, the center of this circle is at the point (2, 3). This means the circle is positioned 2 units to the right of the origin and 3 units up. The center is critical as it serves as the reference point for drawing the circle and locating its position on a coordinate plane.
Radius Calculation
To find the radius of the circle, we look at the right-hand side of the equation. In the standard form \[ (x-h)^{2}+(y-k)^{2}=r^{2} \]\(r^{2}\) is the constant term we compare with our given. For the circle \[ (x-2)^{2}+(y-3)^{2}=4 \]we see that \( r^{2}=4 \).To find \(r\), take the square root of 4, so \[ r = \sqrt{4} = 2 \]Thus, the radius of the circle is 2 units. This radius defines how far all points on the circle are from the center. When graphing, from the center at (2, 3), measure 2 units outwards in all directions to draw the circle accurately.

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Most popular questions from this chapter

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