/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 64 Find the center and the radius o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 64

Find the center and the radius of each circle. Then graph the circle. $$4 x^{2}+4 y^{2}=1$$

Short Answer

Expert verified
Center: (0, 0). Radius: \( \frac{1}{2} \).

Step by step solution

01

Write the equation in standard form

The given equation is \[ 4x^{2} + 4y^{2} = 1 \]Divide every term by 4 to write it in the standard form of a circle鈥檚 equation: \[ x^{2} + y^{2} = \frac{1}{4} \]
02

Identify the center of the circle

The standard form of a circle鈥檚 equation is \[ (x - h)^{2} + (y - k)^{2} = r^{2} \]Here, we have \[ (x - 0)^{2} + (y - 0)^{2} = \frac{1}{4} \]So the center \( (h, k) \) is \( (0, 0) \).
03

Find the radius of the circle

In the standard form, \( r^{2} = \frac{1}{4} \).Taking the square root on both sides, \( r = \frac{1}{2} \).Thus, the radius is \( \frac{1}{2} \).
04

Graph the circle

Plot the center of the circle at \( (0, 0) \).From the center, mark points that are \( \frac{1}{2} \)units away in all directions (up, down, left, and right). Draw a smooth curve connecting these points to create the circle.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Standard Form of Circle
The standard form of a circle's equation helps to identify its center and radius easily. This form is written as \((x - h)^{2} + (y - k)^{2} = r^{2}\), where \(h\) and \(k\) are the coordinates of the circle's center and \(r\) is the radius. In our exercise, the given equation was \[ 4x^{2} + 4y^{2} = 1 \]. By dividing each term by 4, we simplified it to \(x^{2} + y^{2} = \frac{1}{4}\). This matches our standard form with \(h\) and \(k\) both equal to 0 and the radius squared \((r^{2})\) being \(\frac{1}{4}\).
Center of Circle
Locating the center of a circle is straightforward when the equation is in standard form. The general formula \((x - h)^{2} + (y - k)^{2} = r^{2}\) shows that \(h\) and \(k\) are the coordinates of the center. From our simplified equation \(x^{2} + y^{2} = \frac{1}{4}\), we see that \(h\) and \(k\) are both 0. This tells us the center of the circle is at the origin, \( (0, 0) \). Anytime we see \( (x - 0) \) and \( (y - 0) \), the translation to center coordinates is \( (0, 0) \).
Radius of Circle
The radius of a circle can be found by taking the square root of the \(r^{2}\) term in the standard form equation. In \((x - h)^{2} + (y - k)^{2} = r^{2}\), \(r^{2} = \frac{1}{4} \). To find \(r\), take the square root of \(\frac{1}{4} \):
\[ r = \frac{1}{2} \]
This \( \frac{1}{2} \) unit is the distance from the center of the circle to the border in any direction. Thus, the radius of the circle is \(\frac{1}{2}\) units.
Graphing Circles
To graph a circle, follow these simple steps:
  • Identify the center: In our case, it鈥檚 \( (0, 0) \).
  • Mark the center on the graph.
  • Measure the radius distance \( \frac{1}{2} \) units from the center in four directions: up, down, left, and right.
  • Place points at these distances.
  • Draw a smooth curve connecting these points, forming the circle.
Graphing becomes intuitive by practicing these steps: starting at the center, marking equal distances based on the radius, and connecting the dots smoothly to form the circle.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Explain how the equation of a hyperbola differs from the equation of an ellipse.

Find an equation of a hyperbola satisfying the given conditions. Having intercepts \((8,0)\) and \((-8,0)\) and asymptotes \(y=4 x\) and \(y=-4 x\)

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

Solve. $$x^{2}=11$$

Firefighting. The size and shape of certain forest fires can be approximated as the union of two "half-ellipses" 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.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.