/*! 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 55 Find the center and the radius o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 55

Find the center and the radius of each circle. $$ x^{2}+y^{2}=2 $$

Short Answer

Expert verified
The center of the circle is at the origin (0,0) and the radius of the circle is \(\sqrt{2}\).

Step by step solution

01

Identify the center

The equation is already in the standard form i.e., \(x^{2}+y^{2}=r^{2}\). Since there are no constant values added or subtracted with \(x^{2}\) or \(y^{2}\), the center of the circle is at the origin (0,0).
02

Identify the radius

For the standard equation of a circle \(x^{2}+y^{2}=r^{2}\), the right side of the equation gives the square of the radius. In this case, \(r^{2}\) equals 2. To get the radius, take a square root of this value. So, \(r= \sqrt{2}\).

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
In the equation of a circle, the center is a key component that determines its position on a coordinate plane. The most typical format to find this center is in the equation \[(x-h)^2 + (y-k)^2 = r^2\]Here,
  • he center of the circle is \((h, k)\).
If you have an equation like \(x^2 + y^2 = r^2\), which matches the given example \(x^{2}+y^{2}=2\), there are no "h" and "k" values, which means the center is at \((0, 0)\), known as the origin. Understanding this can help one visualize where the circle is plotted even before sketching it.
Radius of a Circle
The radius is the distance from the center of a circle to any point on its circumference. It's crucial in defining the size of a circle, measured as half the diameter. In a circle's equation \((x-h)^2 + (y-k)^2 = r^2\),
  • the radius is denoted by \(r\).
In the provided equation\(x^2 + y^2 = 2\), we see that \(r^2 = 2\). To find the radius itself, taking the square root of 2 is necessary, giving us \(r = \sqrt{2}\). This displays how the radius relates directly to the equation of the circle.
Standard Form of a Circle Equation
The standard form of a circle's equation essentially reveals both the circle's center and its radius. This form is\((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the circle's center, and \(r\) is its radius. Understanding this formulation is critical because:
  • It instantly allows the center and radius of a circle to be human-readable.
  • Simplifies the process of transforming into graphs.
For the equation\(x^2 + y^2 = 2\), it represents a circle with its center at \((0,0)\) and a radius calculated to be \(\sqrt{2}\). MessageLookup the equation will frequently be utilized in geometric and algebraic calculations.
Square Root
The square root is a mathematical operation used to find a number that, when multiplied by itself, yields the original number. For a circle equation like \(x^2 + y^2 = r^2\), finding the radius involves taking the square root of \(r^2\). For \(x^2 + y^2 = 2\), we need \(\sqrt{2}\) to determine the radius.
  • Remember, \(\sqrt{a^2} = a\), so you are effectively "undoing" the squaring process.
  • With non-perfect squares like 2, the square root can't be simplified into an exact integer or simple fraction.
The square root operation provides a direct bridge to understanding dimensions in Euclidean space.
Origin
The origin in a coordinate plane is the point \((0, 0)\). It acts as the central reference point where both the x-axis and y-axis intersect. When a circle's equation simplifies to \(x^2 + y^2 = r^2\), the absence of additional terms involving x and y denotes that:
  • The circle is centered at the origin.
  • Coordinates stand essentially zeroed around the point \((0, 0)\).
Circles centered at the origin make calculating distances, like the radius, straightforward since there's no horizontal or vertical displacement to consider. This foundational knowledge is essential for students dealing with geometric transformations and symmetries.

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 foci for each equation of an ellipse. Then graph the ellipse. $$ \frac{x^{2}}{64}+\frac{y^{2}}{100}=1 $$

Write an equation of an ellipse in standard form with center at the origin and with the given characteristics. focus \((3,0), x\) -intercept \(-6\)

Find the foci for each equation of an ellipse. $$ 25 x^{2}+24 y^{2}=600 $$

Find an equation of an ellipse for each given height and width. Assume that the center of the ellipse is \((0,0) .\) $$ h=5 \mathrm{m}, w=2 \mathrm{m} $$

Which conic section is represented by the equation \(x^{2}+y^{2}=6 x-14 y-9 ?\) F. circle G. ellipse H. parabola J. hyperbola
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Pure 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.