/*! 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 59 Write the standard equation for ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 59

Write the standard equation for each circle. Center at (0,0) with radius \(\sqrt{7}\)

Short Answer

Expert verified
The equation is \ x^2 + y^2 = 7 \

Step by step solution

01

Understand the standard equation of a circle

The standard form of the equation of a circle with center at \(h, k\) and radius \(r\) is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \]
02

Identify the center and radius

Here, the center of the circle is at \(0, 0\) and the radius is \(\frac{\root7}\).
03

Substitute the values into the equation

Substitute \h = 0\, \k = 0\, and \r = \sqrt{7}\ into the standard form of the circle's equation. This gives us: \[ (x - 0)^2 + (y - 0)^2 = (\root7)\^2\]
04

Simplify the equation

Simplify the equation we have: \[ x^2 + y^2 = 7\]

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.

circle equations
A circle's equation in coordinate geometry is a concise mathematical representation that specifies every point on the boundary of the circle. The standard form of a circle's equation is derived from the Pythagorean theorem and is particularly useful in geometry and algebra. The standard form is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \]Here:
  • (x, y) are the coordinates of any point on the circle.
  • (h, k) are the coordinates of the circle's center.
  • r is the radius of the circle.
This equation states that for any point (x, y) to lie on the circle, the distance from that point to the center (h, k) must be equal to the radius r.
center and radius
To write the standard equation of a circle, knowing the center and the radius is crucial. In this exercise, the center is at (0,0), which simplifies our equation significantly. When a circle's center is at (0,0), the equation simplifies from: \[ (x - h)^2 + (y - k)^2 = r^2 \]To:\[ x^2 + y^2 = r^2 \]For this particular circle, the radius (r) is \sqrt[7]. Substituting the radius into the equation, we get: \[ (x - 0)^2 + (y - 0)^2 = (\root 7)^2 \]Which simplifies as: \[ x^2 + y^2 = 7 \]The equation x^2 + y^2 = 7 reflects the relationship between the coordinates of all the points on the circle, the radius, and the center.
simplifying equations
Simplifying equations is a fundamental skill in algebra that helps to make complex mathematical expressions more understandable and easier to work with. In the context of circle equations, simplifying involves substituting known values and reducing the equation to its simplest form. Let's proceed step-by-step:
  • Starting with the standard form of the circle's equation: \[ (x - h)^2 + (y - k)^2 = r^2 \]
  • Identify and substitute the values of (h, k) and r. For our circle, h = 0, k = 0, and r = \root 7.
  • Substitute these into the standard equation: \[ (x - 0)^2 + (y - 0)^2 = (\root 7)^2 \]
  • Simplify \since 0's can be omitted in subtractions: \[ x^2 + y^2 = (\root 7)^2 \]
  • Finally, simplify any remaining terms: \tsince (\root 7)^2 = 7, giving us \[ x^2 + y^2 = 7 \]
Through this process, we've converted the initial detailed representation into a neat, clean standard equation of the circle.

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

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ g(-x) $$

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ f(a) $$

For each pair of variables determine whether \(a\) is a function of \(b\), \(b\) is a function of \(a\), or neither. \(a\) is the length of any piece of U.S. paper currency and \(b\) is its denomination.

Paper Consumption If the average American uses \(580 \mathrm{lb}\) of paper annually, express the total annual paper consumption of the United States, \(P\), as a function of the size of the population, \(n\).

Find the equation of a circle with center (0,1) and radius \(3 .\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

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.