/*! 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 79 How is the standard form of a ci... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 79

How is the standard form of a circle's equation obtained from its general form?

Short Answer

Expert verified
The standard form of a circle's equation, which is \((x-h)^2 + (y-k)^2 = r^2\), is obtained from its general form, \(Ax^2 + By^2 + Cx + Dy + E = 0\), by identifying the coefficients, completing the square and consequently identifying the center \((-C/2, -D/2)\) and the radius \(\sqrt{(C/2)^2 + (D/2)^2 - E}\).

Step by step solution

01

Identify the Coefficients

Firstly, the coefficients of the general form should be well-identified. The equation is given by \(Ax^2 + By^2 + Cx + Dy + E = 0\). Here, A, B, C, D, and E are the coefficients to identify.
02

Convert to Standard Form

Assuming that the coefficients A and B are 1 (since in a circle equation, they must be equal, thus for simplification we take them as 1), the equation can be rewritten as \(x^2 + y^2 + Cx + Dy + E = 0\). The next step to convert this to the standard form \((x-h)^2 + (y-k)^2 = r^2\) is by completing the square. Firstly, group the x terms and the y terms together, giving \(x^2 + Cx + y^2 + Dy + E = 0\). Next, rewrite the equation by completing the square which gives \((x + C/2)^2 - (C/2)^2 + (y + D/2)^2 - (D/2)^2 = -E\). After simplifying, the equation in standard form becomes \((x + C/2)^2 + (y + D/2)^2 = (C/2)^2 + (D/2)^2 - E\).
03

Identify the Center and the Radius

Finally, identify the center and the radius of the circle. Comparing the standard form equation \((x-h)^2 + (y-k)^2 = r^2\) to \((x + C/2)^2 + (y + D/2)^2 = (C/2)^2 + (D/2)^2 - E\), the centre of the circle is identified at \((-C/2, -D/2)\) and the radius is \(\sqrt{(C/2)^2 + (D/2)^2 - E}\).

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

complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$ x^{2}+y^{2}+x+y-\frac{1}{2}=0 $$

Suppose that \(h(x)=\frac{f(x)}{g(x)} .\) The function \(f\) can be even, odd, or neither. The same is true for the function \(g .\) a. Under what conditions is \(h\) definitely an even function? b. Under what conditions is \(h\) definitely an odd function?

determine whether each statement makes sense or does not make sense, and explain your reasoning. Find the area of the donut-shaped region bounded by the graphs of \((x-2)^{2}+(y+3)^{2}=25 \quad\) and \((x-2)^{2}+(y+3)^{2}=36\)

In Exercises \(105-108,\) you will be developing functions that model given conditions. A chemist working on a flu vaccine needs to mix a \(10 \%\) sodium-iodine solution with a \(60 \%\) sodium-iodine solution to obtain a 50 -milliliter mixture. Write the amount of sodium iodine in the mixture, \(S,\) in milliliters, as a function of the number of milliliters of the \(10 \%\) solution used, \(x .\) Then find and interpret \(S(30)\)

use a graphing utility to graph each function. Use \(a[-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$ g(x)=x^{\frac{2}{3}} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.