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Chapter 12: Problem 13

Find the center and radius of each circle. Then graph the circle. $$x^{2}+y^{2}=81$$

Short Answer

Expert verified
The center is (0, 0) and the radius is 9.

Step by step solution

01

Identify the General Circle Equation

Recognize that the given equation, \(x^{2} + y^{2} = 81\), is in the form of a standard circle equation \(x^{2} + y^{2} = r^{2}\). In this form, \(h = 0\) and \(k = 0\) represent the center \((h, k)\), while \(r\) is the radius.
02

Find the Radius

Compare the given equation \(x^{2} + y^{2} = 81\) with the standard form \(x^{2} + y^{2} = r^{2}\). Here, \(r^{2} = 81\). So, solve for \(r\) by taking the square root of both sides: \(r = \sqrt{81}\).
03

Simplify the Radius

Simplify \(r\) to its simplest form: \(r = 9\).
04

Determine the Center of the Circle

The center of the circle \(h, k\) can be determined from the general form of the equation. Since there are no \(h\) and \(k\) values added/subtracted in the equation, the center is \( (0, 0) \).
05

Graph the Circle

To graph the circle, plot the center at the origin \( (0, 0) \), then draw a circle with a radius of 9 units around this center.

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Key Concepts

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

Standard Form of Circle Equation
The standard form of a circle equation is a fundamental concept when learning about circles. It looks like this: \[ (x - h)^2 + (y - k)^2 = r^2 \]. In this equation, the variables \( h \) and \( k \) represent the coordinates of the center of the circle, while \( r \) stands for the radius of the circle. The equation gives you all the information you need to identify and graph a circle. When simplified, it can sometimes appear without \( h \) and \( k \), like in the example \[ x^2 + y^2 = 81 \].
Radius of a Circle
Understanding the radius is key to working with circles. The radius \( r \) is the distance from the center of the circle to any point on its edge. When an equation is in standard form, you can find the radius by taking the square root of the number on the right-hand side of the equation. For example, if the equation is \[ x^2 + y^2 = 81 \], identify \( r \) by solving \[ r = \sqrt{ 81 } \]. This simplifies to \( r = 9 \), giving you the length of the radius.
Center of a Circle
The center of a circle is a foundational concept. In the equation \[ (x - h)^2 + (y - k)^2 = r^2 \], the numbers \( h \) and \( k \) indicate the coordinates of the circle鈥檚 center. They show where the circle is located on the coordinate plane. If there are no \( h \) and \( k \) values, like in the equation \[ x^2 + y^2 = 81 \], the center defaults to \( (0, 0) \). In this specific case, it means the circle is centered at the origin.
Graphing Circles
Graphing a circle involves a few straightforward steps. First, locate the center of the circle on a graph. Then, plot the radius outward from the center in all directions. For instance, with the equation \[ x^2 + y^2 = 81 \], you would graph the center at \( (0, 0) \) and draw a circle extending 9 units out in all directions, because the radius \( r = 9 \). This visual representation helps to understand the circle鈥檚 size and placement within the coordinate system.

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Most popular questions from this chapter

The GL Company makes color television sets. It produces a bargain set that sells for \(\$ 100\) profit and a deluxe set that sells for \(\$ 150\) profit. On the assembly line, the bargain set requires \(3 \mathrm{hr}\), while the deluxe set takes \(5 \mathrm{hr}\). The finishing line spends 1 hr on the finishes for the bargain set and \(3 \mathrm{hr}\) on the finishes for the deluxe set. Both sets require \(2 \mathrm{hr}\) of time for testing and packing. The company has available 3900 work hr on the assembly line, 2100 work hr on the finishing line, and 2200 work hr for testing and packing. How many sets of each type should the company produce to maximize profit? What is the maximum profit?

Suppose that a nonlinear system is composed of equations whose graphs are those described, and the number of points of intersection of the two graphs is as given. Make a sketch satisfying these conditions. (There may be more than one way to do this.) A line and a hyperbola; one point

Which one of the following is a description of the graph of the solution set of the following system? \(x^{2}+y^{2}<25\) \(y>-2\) A. All points outside the circle \(x^{2}+y^{2}=25\) and above the line \(y=-2\) B. All points outside the circle \(x^{2}+y^{2}=25\) and below the line \(y=-2\) C. All points inside the circle \(x^{2}+y^{2}=25\) and above the line \(y=-2\) D. All points inside the circle \(x^{2}+y^{2}=25\) and below the line \(y=-2\)

Solve each system using the elimination method or a combination of the elimination and substitution methods. $$ \begin{array}{r} 2 x^{2}+3 y^{2}=6 \\ x^{2}+3 y^{2}=3 \end{array} $$

Graph each system of inequalities. \(3 x-y>-6\) \(4 x+3 y>12\)
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