/*! 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 47 Find the center and radius of th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 47

Find the center and radius of the circle with equation \(x^{2}+y^{2}-10 x+2 y+\) \(22=0 .\) Then graph the circle.

Short Answer

Expert verified
The circle has center (5, -1) and radius 2.

Step by step solution

01

Write the original equation

Let's start by citing the original equation of the circle: \ \(x^{2}+y^{2}-10x+2y+22=0\).
02

Rearrange the equation

Move the constant term to the right-hand side by subtracting 22 from both sides. This gives: \ \(x^{2} + y^{2} - 10x + 2y = -22\).
03

Complete the square for the x terms

Identify the coefficient of \(x\), which is \(-10\). Take half of it, \(\frac{-10}{2} = -5\), and square it, \((-5)^{2} = 25\). Add and subtract 25 to complete the square: \ \((x^{2} - 10x + 25)\).
04

Complete the square for the y terms

Identify the coefficient of \(y\), which is \(2\). Take half of it, \(\frac{2}{2} = 1\), and square it, \(1^{2} = 1\). Add and subtract 1 to complete the square: \ \((y^{2} + 2y + 1)\).
05

Rewrite the equation in completed square form

Incorporate the completed squares: \ \((x-5)^2 + (y+1)^2 = -22 + 25 + 1\). This simplifies to: \ \((x-5)^2 + (y+1)^2 = 4\).
06

Identify the center of the circle

The equation \((x-5)^2 + (y+1)^2 = 4\) is in the standard form \ \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center. Thus, the center is \((5, -1)\).
07

Identify the radius of the circle

In the equation \((x-5)^2 + (y+1)^2 = 4\), we see that \(r^2 = 4\). Therefore, the radius \(r\) is \(\sqrt{4} = 2\).
08

Graph the circle

To graph the circle, plot the center at \((5, -1)\). Then draw a circle around this center with a radius of 2, ensuring all points are equidistant from the center by 2 units.

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 critical point that defines the circle's position in the coordinate plane. For the equation of a circle \((x - h)^2 + (y - k)^2 = r^2\), the center is represented by the coordinates \((h, k)\). This is where the circle "sits", and all the points on the circle are equidistant from this point.
  • Understanding Coordinate Shift: When looking at the terms \((x - h)\) and \((y - k)\), subtracting a number directly points out the shift from the origin to the new center \((h, k)\).
  • Example: In the problem solved, the equation \( (x-5)^2 + (y+1)^2 = 4 \) tells us the center is at \((5, -1)\).
This makes visualizing the problem easier. Once we know where the center lies, drawing or imagining the circle becomes more straightforward.
Radius of a Circle
The radius of a circle is the distance from the center to any point on the circle's edge. In the equation \((x - h)^2 + (y - k)^2 = r^2\), the radius \(r\) is found by taking the square root of the value on the right side.
  • Mathematical Definition: It is mathematically expressed as \(r = \sqrt{r^2}\).
  • Example: In the given exercise, the right-hand side of the transformed equation is 4, meaning \(r^2 = 4\). This implies that the radius \(r = 2\).
Understanding the radius aids in sketching the circle accurately, ensuring that it captures the correct size and proportions relative to the center.
Completing the Square
Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This technique is indispensable in forming the standard equation of a circle.
  • Process: It involves taking the coefficient of \(x\) or \(y\), dividing it by 2, squaring it, and using this value to adjust the equation.
  • Example: For the \(x\)-terms \(x^2 - 10x\), half of -10 is -5, squaring gives 25. Therefore, the expression becomes \( (x-5)^2\).
This allows us to rewrite the circle equation in a format that reveals the center and the radius directly, simplifying further calculations and interpretations. Completing the square transforms the problem into a manageable form.
Standard Form of a Circle Equation
The standard form of a circle equation is \((x-h)^2 + (y-k)^2 = r^2\), which provides a clear path to identifying the circle's key features like center and radius.
  • Importance: This form makes it easy to quickly see the circle's center and radius.
  • Transformation: Originating from a general quadratic form, the process of rearranging and completing the square converts it to this standard form.
  • Visual Clarity: When plotting, knowing the center \((h, k)\) and radius \(r\) helps you draft a perfect circle in any graphical representation.
In essence, the standard form encapsulates vital information in a concise, neat format, making it crucial for solving problems related to circles.

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 the exact solution(s) of each system of equations. $$ \begin{array}{l}{y^{2}=x^{2}-25} \\ {x^{2}-y^{2}=7}\end{array} $$

HEAIH The prediction equation \(y=205-0.5 x\) relates a person's maximum heart rate for exercise \(y\) and age \(x\) . Use the equation to find the maximum heart rate for an 18 -year old.

Solve each system of inequalities by graphing. $$ \begin{array}{l}{x^{2}+y^{2} \geq 4} \\ {4 y^{2}+9 x^{2} \leq 36}\end{array} $$

REVIEW Given: Two angles are supplementary. One angle is \(25^{\circ}\) more than the measure of the other angle. Conclusion: The measures of the angles are \(65^{\circ}\) and \(90^{\circ} .\) This conclusion \(-\) \(\mathrm{F}\) is contradicted by the first statement given. \(\mathrm{G}\) is verified by the first statement given. H invalidates itself because a \(90^{\circ}\) angle cannot be supplementary to another. J verifies itself because \(90^{\circ}\) is \(25^{\circ}\) more than \(65^{\circ} .\)

Write an equation of a parabola with vertex \((3,1)\) and focus \(\left(3,1 \frac{1}{2}\right) .\) Then draw the graph.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.