Algebra 2
Algebra 2
1st Edition
ISBN: 9780078884825
Author: McGraw-Hill/Glencoe
Publisher: Glencoe/McGraw-Hill School Pub Co
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Concept explainers

Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These…
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are…
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) ge…
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
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Question
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Chapter 10.4, Problem 46HP
To determine

To find: the explanation of the domain and range for an ellipse with equation.

Expert Solution & Answer
Check Mark

Answer to Problem 46HP

The domain and range for an ellipse with equation are {x|−a≤x≤a} and {y|−b≤y≤b}

Explanation of Solution

Given information:

The given equation for an ellipse is x2a2+y2b2=1

Concept used:

Write the standard form of the equation of the ellipse whose orientation is along the horizontal axis is,

  (x−h)2a2+(y−k)2b2=1

The coordinate of the foci is (h±c,k)

The value of c is calculated as,

  c=a2−b2

The vertices are at (h±a,k) and the co-vertices are at (h,k±b) .

Write the standard form of the equation of the ellipse whose orientation is along the vertical axis is,

  (y−k)2a2+(x−h)2b2=1

The coordinate of the foci is (h,k±c)

The vertices are at (h,k±a) and the co-vertices are at (h±b,k) .

The value of c is calculated as,

  c=a2−b2

The given equation of an ellipse is,

  x2a2+y2b2=1    ......(1.)

The equation as a function of x is calculated as,

  1−x2a2=y2b2y2=b2(a2−x2a2)y=±ba(a2−x2)

  y as a function of x is

  y=±ba(a2−x2)      ..... (2.)

The expression in the radical is non-negative,

  a2−x2≥0a2≥x2−a≤x≤a

The domain of an ellipse is {x|−a≤x≤a}

The zero is the minimum value of the radical,

To calculated the value of y to put the minimum value of radical (0) in the function (2.)

  y=±ba(a2−x2)y=±ba×0y=0

To calculate the value of y by putting the maximum value of radical (a) in the equation.(2.)

  y=±ba(a2−x2)y=±ba×ay=±b

The function y lies at between −b and b

The range for an ellipse is {y|−b≤y≤b}

Thus, the domain and range for an ellipse with equation are {x|−a≤x≤a} and {y|−b≤y≤b}

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Chapter 10.4, Problem 45HP
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Chapter 10.4, Problem 47STP

Chapter 10 Solutions

Algebra 2

Ch. 10.1 - Prob. 1ACYPCh. 10.1 - Prob. 1BCYPCh. 10.1 - Prob. 2CYPCh. 10.1 - Prob. 3CYPCh. 10.1 - Prob. 1CYUCh. 10.1 - Prob. 2CYUCh. 10.1 - Prob. 3CYUCh. 10.1 - Prob. 4CYUCh. 10.1 - Prob. 5CYUCh. 10.1 - Prob. 6CYU
Ch. 10.1 - Prob. 7CYUCh. 10.1 - Prob. 8CYUCh. 10.1 - Prob. 9CYUCh. 10.1 - Prob. 10PPSCh. 10.1 - Prob. 11PPSCh. 10.1 - Prob. 12PPSCh. 10.1 - Prob. 13PPSCh. 10.1 - Prob. 14PPSCh. 10.1 - Prob. 15PPSCh. 10.1 - Prob. 16PPSCh. 10.1 - Prob. 17PPSCh. 10.1 - Prob. 18PPSCh. 10.1 - Prob. 19PPSCh. 10.1 - Prob. 20PPSCh. 10.1 - Prob. 21PPSCh. 10.1 - Prob. 22PPSCh. 10.1 - Prob. 23PPSCh. 10.1 - Prob. 24PPSCh. 10.1 - Prob. 25PPSCh. 10.1 - Prob. 26PPSCh. 10.1 - Prob. 27PPSCh. 10.1 - Prob. 28PPSCh. 10.1 - Prob. 29PPSCh. 10.1 - Prob. 30PPSCh. 10.1 - Prob. 31PPSCh. 10.1 - Prob. 32PPSCh. 10.1 - Prob. 33PPSCh. 10.1 - Prob. 34PPSCh. 10.1 - Prob. 35PPSCh. 10.1 - Prob. 36PPSCh. 10.1 - Prob. 37PPSCh. 10.1 - Prob. 38PPSCh. 10.1 - Prob. 39PPSCh. 10.1 - Prob. 40PPSCh. 10.1 - Prob. 41PPSCh. 10.1 - Prob. 42HPCh. 10.1 - Prob. 43HPCh. 10.1 - Prob. 44HPCh. 10.1 - Prob. 45HPCh. 10.1 - Prob. 46HPCh. 10.1 - Prob. 47STPCh. 10.1 - Prob. 48STPCh. 10.1 - Prob. 49STPCh. 10.1 - Prob. 50STPCh. 10.1 - Prob. 51SRCh. 10.1 - Prob. 52SRCh. 10.1 - 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Level curves Consider the paraboloid f(x, y) = 16 x2/4 y,2/16 and the point P on the given level curve of f. ...

Calculus: Early Transcendentals (2nd Edition)

TRY IT YOURSELF 1 Find the mean of the points scored by the 51 winning teams listed on page 39.

Elementary Statistics: Picturing the World (7th Edition)

In Exercises 9-22, change the Cartesian integral into an equivalent polar integral. Then evaluate the polar int...

University Calculus: Early Transcendentals (4th Edition)

CHECK POINT 1 Find a counterexample to show that the statement The product of two two-digit numbers is a three-...

Thinking Mathematically (6th Edition)

the given equation

Pre-Algebra Student Edition

If n is a counting number, bn, read______, indicates that there are n factors of b. The number b is called the_...

Algebra and Trigonometry (6th Edition)

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