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Chapter 13: Problem 15

Find the center and radius of each circle and graph it. $$ x^{2}+y^{2}=9 $$

Short Answer

Expert verified
The center is (0, 0) and the radius is 3.

Step by step solution

01

Identify the Standard Circle Equation Form

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

Compare the Given Equation with the Standard Form

The given equation is \(x^2 + y^2 = 9\). Notice that this equates to \((x - 0)^2 + (y - 0)^2 = 3^2\), implying \(h = 0\), \(k = 0\), and \(r^2 = 9\).
03

Determine the Center of the Circle

From Step 2, the center \((h, k)\) is determined to be \((0, 0)\).
04

Calculate the Radius of the Circle

Since \(r^2 = 9\), taking the square root gives \(r = 3\).
05

Graph the Circle

Plot the center of the circle at the origin \((0, 0)\). Then draw a circle with a radius of 3 units around this point. Ensure the circle extends equally in all directions from the center.

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Key Concepts

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

Standard Form of a Circle's Equation
Understanding the standard form of the equation of a circle is crucial for analyzing circles in algebra. The general expression is \((x - h)^2 + (y - k)^2 = r^2\). This form helps identify key characteristics of a circle like its center and radius. In this expression:
  • \((h, k)\) is the center of the circle. These two values, \(h\) and \(k\), determine the circle's exact location on the Cartesian plane.
  • \(r\) is the radius of the circle, and it is always a positive number as it represents a distance.
When this equation is expanded or altered, recognizing the standard form can help convert it back to reveal the circle's properties. In the given exercise, the formula \(x^2 + y^2 = 9\) can be recognized as having the center at the origin since both \(h\) and \(k\) are zero, simplifying to \((x - 0)^2 + (y - 0)^2 = 3^2\). From this, it is apparent that the radius squared, \(r^2\), is 9.
Radius of a Circle
The radius of a circle is the distance from its center to any point on the circle's edge. It is one of the fundamental properties that defines a circle.
  • In the equation \( (x - h)^2 + (y - k)^2 = r^2 \), the value \(r\) is derived by taking the square root of the term on the right-hand side.
  • For example, if \(r^2 = 9\) as in the given problem, the radius \(r\) is \(\sqrt{9} = 3\).
Calculating the radius is often straightforward, especially with the standard form of a circle's equation. Simply ensure the right side of the equation under the square root remains a positive number, as the radius cannot be negative. This information forms the basis for plotting the circle or determining its size with respect to other components of a graph.
Center of a Circle
The center of a circle provides the precise location around which the circle is constructed. It is point \((h, k)\) in the standard circle equation.
  • The center is crucial because it provides a starting point for drawing the circle.
  • In the given exercise \(x^2 + y^2 = 9\), the structure simplifies to \((x - 0)^2 + (y - 0)^2 = 3^2\), indicating the center is at the origin \((0, 0)\).
Knowing the center helps in understanding the symmetry of the circle and in plotting it correctly on a graph. It is often the reference from which all points on the circle's edge are equidistant, forming the "perimeter" of the circle. Having its coordinates allows you to visualize the circle鈥檚 position in the two-dimensional plane, aiding in broader geometric analyses or comparisons with other geometric shapes.

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Most popular questions from this chapter

On the day of an election, the warning "No electioneering within a \(1,000\) -foot radius of this polling place" was posted in front of a school. Explain what it means.

Solve each system of equations by elimination for real values of x and y. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=25 \\ 2 x^{2}-3 y^{2}=5 \end{array}\right. $$

What happens to the graph of the equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) when \(a=b ?\)

Solve each system of equations for real values of x and y. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=10 \\ 2 x^{2}-3 y^{2}=5 \end{array}\right. $$

Solve each system of equations by substitution for real values of x and y. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=30 \\ y=x^{2} \end{array}\right. $$
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