Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 29: Problem 45
Sketch the indicated curves and surfaces. Sketch the graph of \(x^{2}+y^{2}-2 y=0\) in three dimensions and in two dimensions.
Short Answer
Expert verified
The graph is a circle in 2D, centered at (0,1) with radius 1, and a vertical cylinder in 3D.
Step by step solution
01
Recognize the Type of Equation
The given equation is a quadratic equation in two variables: \(x^2 + y^2 - 2y = 0\). This represents a conic section since it's quadratic in nature.
02
Complete the Square
To simplify and understand the geometry of the equation, complete the square for the \(y\) term:\[x^2 + (y^2 - 2y) = 0 \]Add and subtract 1 inside the parenthesis to complete the square:\[y^2 - 2y + 1 - 1 = (y - 1)^2 - 1\]Substitute back into the equation:\[x^2 + (y - 1)^2 - 1 = 0 \]
03
Rearrange the Equation
Rearrange the equation to identify the standard form of a circle:\[x^2 + (y - 1)^2 = 1 \]This equation represents a circle centered at \((0, 1)\) in the two-dimensional xy-plane with radius 1.
04
Sketch in Two Dimensions
Plot the circle in the xy-plane. The center of the circle is at \((0, 1)\) and the radius is 1 unit. The circle will touch the x-axis at points \((-1, 1)\) and \((1, 1)\).
05
Consider the Graph in Three Dimensions
Although the equation is primarily two-dimensional, if considered in three dimensions (xyz-plane), it can be visualized as a cylinder. The circle from the xy-plane is extended infinitely along the z-axis. This infinite extension forms a cylinder.
06
Draw the Cylinder in Three Dimensions
Visualize (or draw) a vertical cylinder with circular cross-sections in the xy-plane. The circles are all centered along the z-axis, sharing the center at \((0, 1)\) in each xy-plane slice.
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 simplify quadratic equations, allowing us to easily identify key features of conic sections, such as circles. To complete the square, especially for an equation like \(x^2 + y^2 - 2y = 0\), focus on terms involving the same variable, here the \(y\) terms.
Simply put:
In this case, realizing the circle's equation form helps in directly relating it to geometric shapes.
Simply put:
- Look at the \(y^2 - 2y\) portion
- Add and subtract the square of half the coefficient of \(y\) (which is \(-1\))
In this case, realizing the circle's equation form helps in directly relating it to geometric shapes.
Conic Section
Conic sections refer to the curves obtained by intersecting a plane with a cone. These curves include ellipses, circles, parabolas, and hyperbolas. In the context of our equation, \(x^2 + y^2 - 2y = 0\), we are dealing with a circle, a special type of conic section.
Conic sections are important in both mathematics and geometry due to their diverse applications and properties:
Conic sections are important in both mathematics and geometry due to their diverse applications and properties:
- Circles and ellipses are closed curves.
- Parabolas and hyperbolas open infinitely.
Circle
A circle is a collection of points equidistant from a central point. The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
For the equation \(x^2 + (y - 1)^2 = 1\), the circle has:
This simple circle structure is fundamental in understanding more complex geometrical shapes and forms.
For the equation \(x^2 + (y - 1)^2 = 1\), the circle has:
- Center at \((0, 1)\)
- Radius of 1 unit
This simple circle structure is fundamental in understanding more complex geometrical shapes and forms.
Cylinder
A cylinder arises when a circle is extended along a new axis, creating a three-dimensional object with circular cross-sections. In our context, consider the two-dimensional circle \(x^2 + (y-1)^2 = 1\) expanded infinitely along the z-axis.
This geometric approach helps in visualizing three-dimensional figures, especially when considering how an equation might be represented beyond two-dimensional space.
This geometric approach helps in visualizing three-dimensional figures, especially when considering how an equation might be represented beyond two-dimensional space.
- The circle retains its shape in every cross-sectional slice along the z-axis.
- The result is a vertical cylinder with a circular base in the xy-plane.
