/*! 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 60 Write the standard equation for ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 60

Write the standard equation for each circle. Center at (0,0) with radius \(2 \sqrt{3}\)

Short Answer

Expert verified
The standard equation is \ x^2 + y^2 = 12 \.

Step by step solution

01

Recall the Standard Form of Circle Equation

The standard equation for a circle with center \(h, k\) and radius \r\ is given by the formula \( (x - h)^2 + (y - k)^2 = r^2 \).
02

Identify the Center Coordinates

In this problem, the center of the circle is at \(0, 0\), so \(h = 0\) and \(k = 0\).
03

Identify the Radius

The radius \(r\) is given as \( 2 \sqrt{3} \).
04

Substitute Values into the Standard Form

Substitute the center \(h, k\) and the radius \(r\) into the standard form equation: \( (x - 0)^2 + (y - 0)^2 = (2 \sqrt{3})^2 \).
05

Simplify the Equation

Simplify the expression by performing the algebraic operations: \( x^2 + y^2 = 4 \times 3 \).
06

Final Equation

Simplify further to get the final standard equation of the circle: \( x^2 + y^2 = 12 \).

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.

circle equation
The standard equation of a circle is a fundamental concept in geometry, representing a perfect circle on a Cartesian plane.
Understanding this equation is crucial, as it helps in identifying the circle's properties such as its center and radius.
The standard form of the circle equation is given by: \[(x - h)^2 + (y - k)^2 = r^2\]
where \(h\) and \(k\) are the coordinates of the center of the circle, and \(r\) is the circle's radius.
Every point \(x, y\) that satisfies this equation lies on the circle. When you graph such an equation, you get a perfect, round shape.
radius and center
The center and radius are key elements in defining a circle.
The center \(h, k\) is the very middle point of the circle, and the radius \(r\) is the distance from the center to any point on the circle's edge.
In the given exercise, the center is at \(0, 0\), meaning the circle is perfectly centered on the origin of the coordinate system. This simplifies our equation because both \(h\) and \(k\) are zero.
The radius provided is \(2 \sqrt{3}\).
When we substitute the center and radius into the standard form equation, we get: \[(x - 0)^2 + (y - 0)^2 = (2 \sqrt{3})^2\]
which simplifies to: \[x^2 + y^2 = 12\]
algebraic simplification
Algebraic simplification is the process of making an expression more compact and understandable.
In the context of the standard circle equation, once you substitute the center and radius values, you need to simplify the equation for it to retain its standard form.
For example, in our exercise, after substitution, we had: \[(x - 0)^2 + (y - 0)^2 = (2 \sqrt{3})^2\]
Simplifying further:
\[ (x - 0)^2 = x^2 \] and \[ (y - 0)^2 = y^2 \]
Therefore, the expression: \[(x - 0)^2 + (y - 0)^2 = (2 \sqrt{3})^2\] becomes \[ x^2 + y^2 = 4 \times 3 \]
And more simplification leads to: \[ x^2 + y^2 = 12 \]
This is the simplified, standard form equation of our given 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

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ f(3) \cdot k(3) $$

For each pair of variables determine whether \(a\) is a function of \(b\), \(b\) is a function of \(a\), or neither. \(a\) is the time spent studying for the final exam for any student in your class and \(b\) is the student's final exam score.

Graph each function by plotting points and state the domain and range. If you have a graphing calculator, use it to check your results. $$ y=-10-x^{2} $$

Paper Consumption If the average American uses \(580 \mathrm{lb}\) of paper annually, express the total annual paper consumption of the United States, \(P\), as a function of the size of the population, \(n\).

Solve each problem. The number of unmarried-couple households, \(h\) (in millions), can be approximated using the equation $$ h=0.229 n+5.203 $$ where \(n\) is the number of years since 2000 (Census Bureau, www.census.gov). a. Find and interpret the \(n\) -intercept for the line. b. Find and interpret the \(h\) -intercept for the line.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

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.