   Chapter 12.4, Problem 68E ### 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

# DISCUSS: A Family of Confocal Conics Conics that share a focus are called confocal. Consider the family of conics that have a focus at ( 0 , 1 ) and a vertex at the origin, as shown in the figure.(a) Find equations of two different ellipses that have these properties.(b) Find equations of two different hyperbolas that have these properties.(c) Explain why only one parabola satisfies these properties. Find it’s equation.(d) Sketch the conics you found in parts (a),(b), and (c) on the same coordinate axes (for the hyperbolas, sketch top branches only).(e) How are the ellipses and hyperbolas related to the parabola? To determine

a)

The equations of two different ellipses that have the given properties

Explanation

Given:

The family of conics that have a focus at (0,1) and a vertex at the origin.

Approach:

The equation of ellipse is (xh)2b2+(yk)2a2=1.

Here, (h,k) are the coordinates of the center.

2a is the length of major axis.

2b is the length of minor axis.

The distance between center and foci is c=a2b2.

Calculation:

Given one vertex for the ellipse (0,0).

Take other vertex as (0,6).

Figure (1)

Compare the vertex point and foci from Figure (1).

Then, (h,ka)=(0,0),;(h,k+a)=(0,6);(h,kc)=(0,1).

So, h=0,k+a=6,ka=0,kc=1

The values of h,k,a, and c are,

h=0,k=3,a=3,c=2.

Use c=a2b2 to find the value of b.

2=32b24=9b2b=±5

So, the equation of ellipse is x25+(y3)29=1

To determine

b)

The equations of two different hyperbolas that have given properties.

To determine

c)

The equation of a parabola and the reason that only one parabola satisfies these properties.

To determine

d)

To plot:

The conics in part(a), part(b), and part(c) on the same coordinate axes.

To determine

e)

The relation of ellipses and hyperbolas to the parabola.

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