Explanation Given information: The hyperbolas are confocal if its foci are same. Concept used: The equation of a hyperbola for horizontal transverse axis with center ( 0 , 0 ) can be expressed as, x 2 a 2 − y 2 b 2 = 1 ........... (1) The vertices of the hyperbola x 2 a 2 − y 2 b 2 = 1 is ( ± a , 0 ) and foci is ( ± c , 0 ) . The asymptotes of hyperbola x 2 a 2 − y 2 b 2 = 1 is calculated as, y = ± b a x ........... (2) The equation of a hyperbola for vertical transverse axis with center ( 0 , 0 ) can be expressed as, y 2 a 2 − x 2 b 2 = 1 ........... (3) The vertices of the hyperbola y 2 a 2 − x 2 b 2 = 1 is ( 0 , ± a ) and foci is ( 0 , ± c ) . The asymptotes of hyperbola y 2 a 2 − x 2 b 2 = 1 is calculated as, y = ± a b x ........... (4) The relationship between a and c for both equation (1) and equation (3) can be expressed as, c 2 = a 2 + b 2 ........... (5) Proof: The hyperbolas for 0 < k < 16 is, y 2 k − x 2 ( 16 − k ) = 1 To determine (b) To graph: The top branches of family of hyperbolas y 2 k − x 2 ( 16 − k ) = 1 for k = 1 , 4 , 8 , 12 and explain the effect on the shape of graph as k increases.

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Question

Chapter 7.3, Problem 55E

To determine

(a)

To show:

The hyperbolas y2k−x2(16−k)=1 for 0<k<16 are confocal.

To determine

(b)

To graph:

The top branches of family of hyperbolas y2k−x2(16−k)=1 for k=1,4,8,12 and explain the effect on the shape of graph as k increases.

Expert Solution & Answer

Chapter 7 Solutions

College Algebra

Ch. 7.1 - A parabola is the set of all points in the plane...Ch. 7.1 - The graph of the equation x2=4py is a parabola...Ch. 7.1 - The graph of the equation y2=4px is a parabola...Ch. 7.1 - Label the focus, directrix, and vertex on the...Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Graphs of Parabolas Match the equation with the...Ch. 7.1 - Prob. 8E Ch. 7.1 - Prob. 9E Ch. 7.1 - Prob. 10E

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(a) To show: The hyperbolas y 2 k − x 2 ( 16 − k ) = 1 for 0 < k < 16 are confocal.

Solution Summary: The author explains that hyperbolas are confocal if their foci are same.

Videos

(a)

To show:

The hyperbolas y2k−x2(16−k)=1 for 0<k<16 are confocal.

(b)

To graph:

The top branches of family of hyperbolas y2k−x2(16−k)=1 for k=1,4,8,12 and explain the effect on the shape of graph as k increases.

Chapter 7 Solutions

(a) To show: The hyperbolas y 2 k − x 2 ( 16 − k ) = 1 for 0 &lt; k &lt; 16 are confocal.

Solution Summary: The author explains that hyperbolas are confocal if their foci are same.

Videos

(a) To show: The hyperbolas y2k−x2(16−k)=1 for 0<k<16 are confocal.

(b) To graph: The top branches of family of hyperbolas y2k−x2(16−k)=1 for k=1,4,8,12 and explain the effect on the shape of graph as k increases.

Chapter 7 Solutions

(a) To show: The hyperbolas y 2 k − x 2 ( 16 − k ) = 1 for 0 < k < 16 are confocal.

(a)

To show:

The hyperbolas y2k−x2(16−k)=1 for 0<k<16 are confocal.

(b)

To graph:

The top branches of family of hyperbolas y2k−x2(16−k)=1 for k=1,4,8,12 and explain the effect on the shape of graph as k increases.