/*! 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 66 Find the center and radius of th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 66

Find the center and radius of the circle. $$ (x-5)^{2}+y^{2}=19 $$

Short Answer

Expert verified
Center: (5, 0); Radius: \(\sqrt{19}\).

Step by step solution

01

Identify the Standard Form of a Circle Equation

The equation of a circle in the standard form is given by \((x-h)^2 + (y-k)^2 = r^2\) where \(h, k\) is the center of the circle and \(r\) is the radius. In this form, \(h\) and \(k\) are the x and y coordinates of the circle's center, respectively.
02

Compare Given Equation with Standard Form

The given equation is \(x - 5\)^2 + y^2 = 19\. Comparing this with the standard form, we find that \(h = 5\) and \(k = 0\). Thus, the center of the circle is \(h, k = (5, 0)\).
03

Find the Radius of the Circle

From the standard form equation, \(r^2 = 19\). Therefore the radius \(r\) is found by taking the square root of 19, that is \( r = \sqrt{19}.\)
04

State the Solution

The center of the circle is at (5, 0) and the radius of the circle is \(\sqrt{19}\).

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.

Center of a Circle
The center of a circle is a crucial point that defines its exact location in a plane. Think of it as the circle's "heart," from which every point on the boundary is equidistant.
To find the center from an equation in the standard form,
  • look at the values of \(h\) and \(k\) in \((x-h)^2 + (y-k)^2 = r^2\).
In our example, the expression \((x-5)^2+y^2=19\) shows us that \(h=5\) and \(k=0\).
Thus, the center is located at the point (5, 0).
This point is key to understanding the circle's position and helping us plot it accurately on a graph.
Radius of a Circle
The radius of a circle is the distance from its center to any point on the edge, consistently the same all around.
In the standard form of the circle's equation, this is represented by \(r\) in
  • \((x-h)^2 + (y-k)^2 = r^2\).
From our equation, \(r^2 = 19\), so the radius \(r\) is simply \(\sqrt{19}\).
Understanding the radius is key to determining the size of the circle.
Although \(\sqrt{19}\) is not an integer, it tells us how large the circle is and how far points on its boundary are from its center.
Standard Form of a Circle
The standard form of a circle is an algebraic way to represent all the points that make up the circle. It makes understanding and identifying the circle's key features easy.
The general format is:
  • \((x-h)^2 + (y-k)^2 = r^2\)
Here, \((h, k)\) represents the coordinates of the circle's center, and \(r\) signifies the radius.
By comparing any circle equation to this form, you can quickly extract useful information about its center and size. This makes it an invaluable tool in both geometry and algebra, enabling students to transition easily between equations and visual representations.

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 you could use a complete numerical representation (table) for a function to determine its domain and range.

Going Green The average top-loading washing machine uses about 40 gallons of water per load of clothes. Write the formula for a function \(W\) that calculates the number of gallons of water used while washing \(x\) loads of clothes. Evaluate \(W(30)\) and interpret your result.

Calculate the percent change for the given A and B. Round your answer to the nearest tenth of a percent when appropriate. $$ A=1256, B=1195 $$

Suppose you are given a graphical representation of a function \(f .\) Explain how you would determine whether \(f\) is constant, linear, or nonlinear. How would you determine the type if you were given a numerical or symbolic representation? Give examples.

Determine if \(S\) is a function. $$ S=\\{(a, 2),(a, 3),(b, 5),(-b, 7)\\} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.