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Chapter 13: Problem 41

Find the center and the radius of each circle. Then graph the circle. $$x^{2}+y^{2}=64$$

Short Answer

Expert verified
Center: (0, 0); Radius: 8. Plot the circle centered at the origin with radius 8.

Step by step solution

01

Identify the general form of the circle equation

The given equation is in the form of \[ x^{2}+y^{2}=64 \].
02

Convert to standard form

The equation \( x^{2}+y^{2}=64 \) is already in the standard form of a circle equation, which is \( (x - h)^{2} + (y - k)^{2} = r^{2} \).
03

Identify the center

In the standard form \((x - h)^{2} + (y - k)^{2} = r^{2} \), the center \((h, k) \) can be identified. Here, \( h = 0 \) and \( k = 0 \), so the center of the circle is \((0,0)\).
04

Identify the radius

In the standard form \((x - h)^{2} + (y - k)^{2} = r^{2} \), the radius can be found by taking the square root of the constant term on the right side: \( r^{2} = 64 \) thus \( r = \sqrt{64} = 8 \).
05

Graph the circle

To graph the circle with its center at \((0, 0)\) and a radius of \(8\), plot the center point at the origin \((0, 0)\). Then, draw a circle with a radius of \(8\) units around the center.

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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 point that determines its position in the coordinate plane. It's denoted as \((h, k)\) in the standard form of a circle equation \((x - h)^2 + (y - k)^2 = r^2\). In this form, \h\ and \k\ represent the x and y coordinates of the center, respectively.

A circle is perfectly symmetrical around its center. Knowing the center helps in graphing the circle accurately.

For example, in the equation \x^2 + y^2 = 64\, if you compare it to the standard form, you can see that both \h\ and \k\ are zero. Hence, the center of the circle is at the origin, \((0, 0)\).
radius
The radius of a circle is the distance from the center to any point on the circle. In the standard form of a circle equation \((x - h)^2 + (y - k)^2 = r^2\), the radius is represented by \r\.

To find the radius, you take the square root of the number on the right side of the equation. For instance, in the equation \x^2 + y^2 = 64\, the number on the right side is 64. Taking the square root of 64, we get 8. Thus, the radius of the circle is 8 units.

Understanding the radius is important as it helps in drawing the circle around its center, ensuring it maintains its perfect round shape.
standard form of a circle
The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\). This form is very helpful for quickly identifying the key features of a circle, specifically its center and radius.

In this equation:
  • \ (h, k) \ is the center of the circle.
  • \ r \ is the radius of the circle, which is the square root of the number on the right side of the equation.
For example, the equation \x^2 + y^2 = 64\ is in standard form, making it easy to see that:
  • The center is at \ (0, 0) \, because there are no \ h \ and \ k \ values subtracted from \x\ and \y\.
  • The radius is 8, because \ \sqrt{64} = 8 \.
By practicing with the standard form, you can quickly graph and understand any given circle.

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Most popular questions from this chapter

Match the equation with the center or vertex of its graph, listed in the column on the right. a) Vertex: \((-2,5)\) b) Vertex: \((5,-2)\) c) Vertex: \((2,-5)\) d) Vertex: \((-5,2)\) e) Center: \((-2,5)\) f) Center: \((2,-5)\) g) Center: \((5,-2)\) h) Center: \((-5,2)\) $$y=(x-5)^{2}-2$$

Match the equation with the center or vertex of its graph, listed in the column on the right. a) Vertex: \((-2,5)\) b) Vertex: \((5,-2)\) c) Vertex: \((2,-5)\) d) Vertex: \((-5,2)\) e) Center: \((-2,5)\) f) Center: \((2,-5)\) g) Center: \((5,-2)\) h) Center: \((-5,2)\) $$(x+2)^{2}+(y-5)^{2}=9$$

Find an equation of a circle satisfying the given conditions. The endpoints of a diameter are \((7,3)\) and \((-1,-3)\)

Firefighting. The size and shape of certain forest fires can be approximated as the union of two "halfellipses." For the blaze modeled below, the equation of the smaller ellipse - the part of the fire moving into the wind- is $$ \frac{x^{2}}{40,000}+\frac{y^{2}}{10,000}=1 $$ The equation of the other ellipse - the part moving with the wind- is $$ \frac{x^{2}}{250,000}+\frac{y^{2}}{10,000}=1 $$ Determine the width and the length of the fire. Source for figure: "Predicting Wind-Driven Wild Land Fire Size and Shape," Hal E. Anderson. Rescarch Paper INT-305. U.S. Department of Agriculture, Forest Service, February 1983

Solve $$ \begin{aligned} &a+b=\frac{5}{6}\\\ &\frac{a}{b}+\frac{b}{a}=\frac{13}{6} \end{aligned} $$
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