   Chapter 12.4, Problem 52E ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

#### Solutions

Chapter
Section ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

# 47-58 Graphing Shifted Conics Complete the square to determine whether the graph of the equation is an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and the lengths of the major and minor axes. If it is parabola, find the vertex, focus, and directrix. If it is hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. if the equation has no graph, explain why. 2 x 2 + y 2 = 2 y + 1 .

To determine

The graph of the equation and its characteristics.

Explanation

Given:

The given equation is,

2x2+y2=2y+1

Approach:

The basic equation for an ellipse with a vertical major axis which is used is as follows,

x2b2+y2a2=1

Where a and b are constant values.

Expression to find vertices is,

V=(0,±a)

Expression to find Foci is,

F=(0,±c),

Where, c2=a2b2

Calculation:

Consider the given equation,

2x2+y2=2y+1

Now complete the square as shown below,

2x2+y22y=12x2+y22y+1=1+12x2+(y1)2=22x22+(y1)22=1

Further solve the above equation.

x21+(y1)22=1

This is an equation of a shifted ellipse with center (0,1).

Here,

a2=2

And,

b2=1

It is obtained from the equation of the ellipse at the center,

x2a2+y2b2=1

That is,

x21+y22=1

The end points of the minor and major axes of the un-shifted ellipse are,

(1,0),(1,0),(0,2) and (0,2).

The end points of the minor and major axes of the shifted ellipse are,

(1,0)(1+0,0+1)=(1,1)(1,0)(1+0,0+1)=(1,1)(0,2)(0+0,2+1)=(0,1+2)(0,2)(0+0,2+1)=(0,12)

Here,

a2=2b2=1

Then,

c2=a2b2c2=21c2=1c=±1

Thus the un-shifted ellipse foci are,

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