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Chapter 13: Problem 11

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

Short Answer

Expert verified
Center: (0, 0), Radius: 2

Step by step solution

01

- Identify the Equation Form

The given equation is \(x^2 + y^2 = 4\). This is a standard form of the circle's equation which is \(x^2 + y^2 = r^2\).
02

- Determine the Center

For the equation \(x^2 + y^2 = r^2\), the center of the circle is at the origin, \( (0, 0) \).
03

- Find the Radius

Compare the given equation \(x^2 + y^2 = 4\) to the standard form \(x^2 + y^2 = r^2\). Here \(r^2 = 4\), thus the radius \(r = \sqrt{4} = 2\).
04

- Graph the Circle

To graph the circle, plot the center at \( (0, 0) \). Then draw a circle with radius 2 units from the origin in all directions.

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Key Concepts

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

circle center
A circle's center is a crucial point that defines its shape and position. In the given equation, \(x^2 + y^2 = 4\), we have a perfect example of a circle in standard form. The standard form of a circle equation is \(x^2 + y^2 = r^2\). Here, there are no additional terms with x or y, indicating that the circle is centered at the origin.
In other words, the coordinates of the center are (0,0). This makes it simple to locate the circle's position on a graph.
circle radius
The radius of a circle is the distance from the center to any point on the circle. It plays a key role in determining the size of the circle. To find the radius from the equation \(x^2 + y^2 = r^2\), we compare it to the given equation \(x^2 + y^2 = 4\). Here, \(r^2 = 4\), so to find the radius, we take the square root of 4:
  • \( r = \sqrt{4} \)
  • \( r = 2 \)
This means the radius of the circle is 2 units. Knowing the radius helps us to graph the circle accurately.
graphing circles
Graphing a circle involves plotting its center and using the radius to draw the circle accurately. For the equation \(x^2 + y^2 = 4\), we know the center is at (0,0) and the radius is 2 units.
Follow these steps to graph the circle:
  • Plot the center at the origin, point (0,0).
  • Using a compass or a steady hand, measure 2 units outward from the center in all directions.
  • Draw the circle, ensuring all points are exactly 2 units away from the center.
Now you have a clear, accurately drawn circle on your graph.

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Most popular questions from this chapter

Solve each system using the elimination method or a combination of the elimination and substitution methods. $$ \begin{array}{l} x^{2}+y^{2}=41 \\ x^{2}-y^{2}=9 \end{array} $$

Graph each system of inequalities. \(y \leq-x^{2}\) \(y \geq x-3\) \(y \leq-1\) \(x<1\)

Graph each function. Give the domain and range. \(f(x)=|x+1|\)

Solve each system using the substitution method. $$ \begin{array}{l} 2 x^{2}+4 y^{2}=4 \\ x=4 y \end{array} $$

Evaluate each expression. [-14]
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