/*! 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 77 What is a circle? Without using ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 77

What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.

Short Answer

Expert verified
A circle is defined as a collection of points equidistant from a single point, the center. The equation of a circle can be derived from its definition, resulting in: (X-a)^2 + (Y-b)^2 = R^2, where (a,b) is the center of the circle, and R is the radius.

Step by step solution

01

Define a Circle

A circle can be defined as a set of all points in a plane that are equidistant from a fixed point called the center.
02

Explain the Distance From the Center to Any Point on the Circle

The fixed distance from the center of the circle to any point on the circle is called the radius. Every point on the circle's circumference is exactly the same distance (the radius) from the circle's center.
03

Deriving the Circle's Equation

The definition of the circle gives us a way to formulate the equation of a circle. Imagine a circle centered at origin. Any point (X,Y) on the circle meets the condition that it's a distance 'R' away from the center (0,0). The distance between two points is determined by the Pythagorean Theorem in the Cartesian plane. Here, since the distance is 'R', we have 'R' equals to the square root of (X^2 + Y^2), which, when squared, results in the equation of a circle: X^2 + Y^2 = R^2. Now, If the center is not at the origin (0,0), but at (a, b), the expression of distance changes slightly, resulting in the alternate form of the circle's equation: (X-a)^2 + (Y-b)^2 = R^2.

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!

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

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=7$$

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}-2 x+y^{2}-15=0$$

Show that the points \(A(1,1+d), B(3,3+d),\) and \(C(6,6+d)\) are collinear (lie along a straight line) by showing that the distance from \(A\) to \(B\) plus the distance from \(B\) to \(C\) equals the distance from \(A\) to \(C\).

Determine whether each statement makes sense or does not make sense, and explain your reasoning.I used a function to model data from 1990 through 2015 .I have two functions. Function \(f\) models total world population \(x\) years after 2000 and function \(g\) models population of the world's more-developed regions \(x\) years after 2000.1 can use \(f-g\) to determine the population of the world's less-developed regions for the years in both function's domains.

Determine whether each statement makes sense or does not make sense, and explain your reasoning.I used a function to model data from 1990 through 2015 . The independent variable in my model represented the number of years after \(1990,\) so the function's domain was \(\\{x | x=0,1,2,3, \dots, 25\\}\).
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Calculus

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.