/*! 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 31 Write the center-radius form of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 31

Write the center-radius form of each circle described. Then graph the circle. $$ x^{2}+y^{2}+10 x-14 y-7=0 $$

Short Answer

Expert verified
The center-radius form is \[(x + 5)^2 + (y - 7)^2 = 81\]. The center is \((-5, 7)\) and the radius is 9.

Step by step solution

01

- Write the general form of the circle equation

Identify the given equation of the circle and write it in standard form. Currently, the given equation is \[ x^{2} + y^{2} + 10x - 14y - 7 = 0 \].
02

- Group x and y terms

Reorganize the equation to group the x and y terms together: \[ (x^{2} + 10x) + (y^{2} - 14y) = 7 \].
03

- Complete the square for x terms

To complete the square for the x-terms, use the formula \[(x + \frac{b}{2})^2 = x^2 + bx + \frac{b^2}{4}\]. For the x-terms (x^2 + 10x), add and subtract \[(\frac{10}{2})^2 = 25\]: \[ (x^2 + 10x + 25 - 25) \].
04

- Complete the square for y terms

Similarly, for the y-terms \[(y^2 - 14y + 49 - 49)\]: \[ (y + \frac{-14}{2})^2 = y^2 - 14y + 49 \].
05

- Rewrite the equation

Rewrite the equation by replacing the completed square forms: \[ (x + 5)^2 - 25 + (y - 7)^2 - 49 = 7 \].
06

- Simplifying

Simplify the equation to standard form: \[ (x + 5)^2 + (y - 7)^2 - 25 - 49 = 7 \] which simplifies to \[ (x + 5)^2 + (y - 7)^2 = 81 \].
07

- Identify center and radius

Compare this with the standard form \[(x - h)^2 + (y - k)^2 = r^2\]. This gives the center \((h, k) = (-5, 7)\) and the radius \(r = \sqrt{81} = 9\).
08

- Graph the circle

To graph the circle, plot the center at \((-5, 7)\) and draw a circle with radius 9 units around this center.

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.

completing the square
Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This is especially useful for converting the general form of a circle's equation into the center-radius form. To complete the square:
  • First, group x and y terms together and move the constant to the other side of the equation.
  • For each group, add and subtract the square of half the linear coefficient. This keeps the equation balanced.
For example, to complete the square for the term \(x^2 + 10x\), we take half of 10, square it, and then add and subtract 25:
\(x^2 + 10x + 25 - 25\) which rewrites as \((x + 5)^2 - 25\).
Repeat for the y terms:
\(y^2 - 14y + 49 - 49\) which simplifies to \((y - 7)^2 - 49\).
This technique transforms the given equation into a more manageable form that reveals the circle's center and radius.
center-radius form
The center-radius form of a circle's equation is presented as:

\((x - h)^2 + (y - k)^2 = r^2\)

where:
  • \((h, k)\) is the center of the circle.
  • \(r\) is the radius.
Converting our example, we start by completing the square for both x and y terms. After performing the steps:
\((x + 5)^2 + (y - 7)^2 = 81\)
Aligning this with the center-radius form, we identify the circle's center at \((-5, 7)\) and the radius as \(\sqrt{81} = 9\).
This clear form makes it straightforward to understand and graph a circle.
graphing circle
Graphing a circle in the center-radius form is straightforward. Follow these steps:
  • Identify the center \(h, k\).
  • Identify the radius \(r\).
  • Plot the center on the coordinate plane.
  • Use the radius to mark several points around the center, ensuring they are all exactly \(r\) units away.
In our example, the center is \((-5, 7)\) and the radius is 9 units. To graph the circle:
Plot the center point at \((-5, 7)\). Mark points 9 units in all directions: left, right, up, and down from the center. Connect these points smoothly to form a circle. Using a compass or a flexible curve tool can make it easier to draw an accurate circle. Always check your points to ensure the distances are consistent.

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

Solve each system using the elimination method or a combination of the elimination and substitution methods. $$ \begin{array}{l} 2 x^{2}=8-2 y^{2} \\ 3 x^{2}=24-4 y^{2} \end{array} $$

Solve each system using the substitution method. $$ \begin{array}{l} x^{2}+y^{2}=2 \\ 2 x+y=1 \end{array} $$

Write the center-radius form of each circle described. Then graph the circle. $$ x^{2}+y^{2}-8 x-12 y+3=0 $$

Solve each system using the elimination method or a combination of the elimination and substitution methods. $$ \begin{array}{r} x^{2}+6 y^{2}=9 \\ 4 x^{2}+3 y^{2}=36 \end{array} $$

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

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Applied 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.