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Chapter 10: Problem 144

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

Short Answer

Expert verified
The center of the circle is (0,0) and the radius is 2.

Step by step solution

01

- Identify Equation Form

Recognize the standard form for a circle's equation which is ** Identify the current equation form **
02

- Rewrite the Equation in Standard Form

The provided equation is already in the standard form: x y => equation
03

- Determine the Center

The standard form shows that the center as form center form .
04

- Determine the Radius

The right-hand side of the value, Find and radius.
05

- Graph the Circle

Draw a coordinate plane and plot the center of the circle. Identify points on the plane that are a distance equal to the radius from the center and sketch the circle.

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Key Concepts

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

center of a circle
The center of a circle is a crucial concept when graphing circles. It signifies the exact middle point of the circle, from which all points on the circumference are equidistant. In the equation presented, the standard form of a circle is \( (x-h)^2 + (y-k)^2 = r^2 \). Here, \text{(h, k)}\ is the center of the circle.

For the given problem, the equation is \(x^{2} + y^{2} = 4\). This implies \(h = 0\) and \(k = 0\), putting the center at the origin \text{(0, 0)}\.

Understanding the center allows us to correctly plot the circle on a coordinate plane.
  • The center is the starting point for determining the circle's radius.
  • It ensures the circle is symmetrical and correctly positioned on the graph.
radius of a circle
The radius of a circle is the distance from the center to any point on the circle's edge. It is a key piece of information for sketching the circle accurately. In the standard form of a circle's equation \( (x-h)^2 + (y-k)^2 = r^2 \), the term \( r^2 \) represents the square of the radius.

For the current equation, \(x^{2} + y^{2} = 4\), we can see that \( r^2 = 4\). By taking the square root of both sides, we find that \( r = 2\).

The radius is essential to:
  • Ensure the accuracy of the circle's size on your graph.
  • Locate the points around the center that define the circle's edge.
standard form of circle equation
The standard form of a circle's equation is central to understanding how to graph and analyze circles. This form is \( (x-h)^2 + (y-k)^2 = r^2 \), where:
  • \text{(h, k)} is the center of the circle.
  • r is the radius of the circle.


To graph a circle using this equation:
  • Identify the center (\text{h, k}) and the radius (r).
  • Plot the center point on a coordinate plane.
  • Measure out distance 'r' from the center in all directions to sketch the circle accurately.
For the given equation \(x^{2} + y^{2} = 4\), it perfectly matches the standard form as \( (x-0)^2 + (y-0)^2 = 2^2 \). This directly shows the center is at \text{(0, 0)}\text{ and the radius is 2}. Understanding and using the standard form of the circle equation simplifies the graphing process and enhances comprehension of geometric properties.

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Most popular questions from this chapter

Simplify \(-\sqrt[4]{1250}\)

Simplify. Assume that \(x \geq 0\) \(\sqrt[12]{x^{38}}\)

Work each problem. Suppose someone claims that \(\sqrt[n]{a^{n}+b^{n}}\) must equal \(a+b,\) because when \(a=1\) and \(b=0,\) a true statement results: $$ \sqrt[n]{a^{n}+b^{n}}=\sqrt[n]{1^{n}+0^{n}}=\sqrt[n]{1^{n}}=1=1+0=a+b $$ Explain why this is faulty reasoning.

Graph each circle. Identify the center and the radius. \(x^{2}+y^{2}=16\)

Find the distance between each pair of points. (6,13) and (1,1)
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