/*! 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 46 State the center and radius of t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 46

State the center and radius of the circle with the given equations. $$(x+3)^{2}+(y-7)^{2}=81$$

Short Answer

Expert verified
The center of the circle is \((-3, 7)\) and the radius is 9.

Step by step solution

01

Identify the Standard Equation of a Circle

Recall that the standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) represents the center of the circle and \(r\) is the radius.
02

Compare Given Equation with Standard Form

Given the equation \((x + 3)^2 + (y - 7)^2 = 81\), compare this with the standard form \((x - h)^2 + (y - k)^2 = r^2\). Identify \(h\) and \(k\) from the equation.
03

Determine the Center of the Circle

For \((x + 3)^2\), rewrite it as \((x - (-3))^2\). This shows that \(h = -3\). For \((y - 7)^2\), \(k = 7\). Thus, the center of the circle is \((-3, 7)\).
04

Solve for the Radius of the Circle

The equation has \(r^2 = 81\), so the radius \(r\) is the square root of 81. Calculate \(r = \sqrt{81} = 9\).
05

State Final Answer

Combining what we have, the center of the circle is \((-3, 7)\) and the radius is 9.

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.

Standard Form of a Circle
In geometry, the equation of a circle is crucial for identifying its properties, and the standard form makes this task simple. The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\). Understanding this structure is key:

  • \((x - h)\) and \((y - k)\) are the algebraic expressions that help locate the center of the circle. Reversal of signs happens here to find the actual coordinates of the center.
  • \(r^2\) signifies the square of the circle's radius, meaning you need to find the square root to determine the actual radius length.
Using the standard form helps in straightforward comparisons with given equations and makes it easy to extract the needed details about a circle鈥檚 geometry.
Center of a Circle
The center of a circle, denoted as \((h, k)\), is found by carefully analyzing the given equation. The sense of direction is reversed from how the equation is written to find the center's actual coordinates. For example, if the equation is given as \((x + 3)^2 + (y - 7)^2 = 81\), one might initially think the center is \((3, 7)\).


However, closer inspection reveals that these expressions represent \((x - (-3))\) and \((y - 7)\). This indicates the coordinates are actually \((-3, 7)\).

It's essential to remember:
  • The signs of \(h\) and \(k\) will be the opposite of what appears in the equation.
  • Rewriting expressions can make understanding easier.
Grasping how to find the center empowers you to navigate other features of the circle, such as radius calculations.
Radius of a Circle
The radius of a circle is the constant distance from the center to any point on the circle. In standard form equations, the term \(r^2\) is given, where \(r\) represents the radius itself. To find the radius, you simply take the square root of \(r^2\).

For instance, if an equation presents as \(r^2 = 81\), the radius \(r\) is calculated by \(\sqrt{81}\), resulting in \(9\).

Therefore, identifying the radius involves two key steps:
  • Spot \(r^2\) in the equation.
  • Compute the square root.
The radius is a fundamental dimension of the circle, affecting its size and properties, and is pivotal in any geometric or algebraic analysis.

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

Temperature. According to NOAA, a "standard day" is \(15^{\circ} \mathrm{C}\) at sea level, and every 500 -feet elevation above sea level corresponds to a 1 ' \(\mathrm{C}\) temperature drop. Assuming the relationship between temperature and elevation is linear, write an equation that models this relationship. What is the expected temperature at 2500 feet on a "standard day"?

Determine whether the following statements are true or false. The quadratic formula can be used to solve any quadratic equation.

Monthly Driving Costs. The monthly costs associated with driving a new Honda Accord are the monthly loan payment plus \(\$ 25\) every time you fill up with gasoline. If you fill up 5 times in a month, your total monthly cost is \(\$ 500 .\) How much is your loan payment?

Solve for the indicated variable in terms of other variables. Solve \(A=P(1+r)^{2}\) for \(r\)

Write a quadratic equation in standard form that has the solution set \(\\{-3,0\\} .\) Alternate solutions are possible.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Decision Maths

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.