Explanation The given equation of the hyperbola is x 2 a 2 − y 2 b 2 = 1 . For the given equation, the vertices are ( a , 0 ) and ( − a , 0 ) , end of conjugate axis are ( 0 , b ) and ( 0 , − b ) as shown in figure 1. Fundamental rectangle of a hyperbola is formed by the help of vertices ( V and V’ ) and ends of conjugate axis of a hyperbola ( B and B’ ). Vertices and ends of conjugate axis are the mid points of the sides of fundamental rectangle. The rectangle R S P Q is the fundamental rectangle of the hyperbola x 2 a 2 − y 2 b 2 = 1 as shown is figure 1. Diagonals of the fundamental rectangle are R P and S Q . The coordinates of the vertex R for a fundamental rectangle R S P Q is ( − a , b ) . It is obtained by the symmetry of the rectangle R S P Q along each axis and observing the vertex V ' ( − a , 0 ) and end of conjugate axis B ( 0 , b ) . Similarly, coordinates of the vertex P for a fundamental rectangle R S P Q is ( a , − b ) . It is obtained by observing the vertex V ( a , 0 ) and end of conjugate axis B ' ( 0 , − b ) . Coordinates of the vertex S for a fundamental rectangle R S P Q is ( a , b ) . It is obtained by observing the vertex V ( a , 0 ) and end of conjugate axis B ( 0 , b ) . Coordinates of the vertex Q for a fundamental rectangle R S P Q is ( − a , − b ) . It is obtained by observing the vertex V ' ( − a , 0 ) and end of conjugate axis B ' ( 0 , − b ) . Now, obtain the equation of a line that passes through the diagonal R P by using the formula ( y − y 1 ) = ( y 1 − y 2 ) ( x 1 − x 2 ) ( x − x 1 ) , where ( x 1 , y 1 ) and ( x 2 , y 2 ) are two points on a line. For the equation of a line that passes through the diagonal points R ( − a , b ) and P ( a , − b ) , substitute ( x 1 , y 1 ) ≡ ( − a , b ) and ( x 2 , y 2 ) ≡ ( a , − b ) in the equation ( y − y 1 ) = ( y 1 − y 2 ) ( x 1 − x 2 ) ( x − x 1 ) as follows:

12th Edition

ISBN: 9781305652231

Author: R. David Gustafson, Jeff Hughes

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 7.3, Problem 70E

To determine

To Show:

Extended diagonals of the fundamental rectangle of the hyperbola x2a2−y2b2=1 are y=bax and y=−bax.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.3, Problem 69E

Chapter 7.3, Problem 71E

Students have asked these similar questions

Assume {u1, U2, us} spans R³. Select the best statement. A. {U1, U2, us, u4} spans R³ unless u is the zero vector. B. {U1, U2, us, u4} always spans R³. C. {U1, U2, us, u4} spans R³ unless u is a scalar multiple of another vector in the set. D. We do not have sufficient information to determine if {u₁, u2, 43, 114} spans R³. OE. {U1, U2, 3, 4} never spans R³. F. none of the above

Assume {u1, U2, 13, 14} spans R³. Select the best statement. A. {U1, U2, u3} never spans R³ since it is a proper subset of a spanning set. B. {U1, U2, u3} spans R³ unless one of the vectors is the zero vector. C. {u1, U2, us} spans R³ unless one of the vectors is a scalar multiple of another vector in the set. D. {U1, U2, us} always spans R³. E. {U1, U2, u3} may, but does not have to, span R³. F. none of the above

Let H = span {u, v}. For each of the following sets of vectors determine whether H is a line or a plane. Select an Answer u = 3 1. -10 8-8 -2 ,v= 5 Select an Answer -2 u = 3 4 2. + 9 ,v= 6

Chapter 7 Solutions

College Algebra (MindTap Course List)

Ch. 7.1 - Prob. 1SC Ch. 7.1 - Prob. 2SC Ch. 7.1 - Prob. 3SC Ch. 7.1 - Prob. 4SC Ch. 7.1 - Prob. 5SC Ch. 7.1 - Prob. 6SC Ch. 7.1 - Find the vertex and yintercepts of the parabola...Ch. 7.1 - Prob. 8SC Ch. 7.1 - Prob. 9SC Ch. 7.1 - Prob. 1E

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

Extended diagonals of the fundamental rectangle of the hyperbola x 2 a 2 − y 2 b 2 = 1 are y = b a x and y = − b a x .

Solution Summary: The author explains the fundamental rectangle of the hyperbola, which is formed by the help of vertices and ends of conjugate axis.

Videos

To Show:

Extended diagonals of the fundamental rectangle of the hyperbola x2a2−y2b2=1 are y=bax and y=−bax.

Want to see the full answer?

Chapter 7 Solutions

Extended diagonals of the fundamental rectangle of the hyperbola x 2 a 2 − y 2 b 2 = 1 are y = b a x and y = − b a x .

Solution Summary: The author explains the fundamental rectangle of the hyperbola, which is formed by the help of vertices and ends of conjugate axis.

Videos

To Show: Extended diagonals of the fundamental rectangle of the hyperbola x2a2−y2b2=1 are y=bax and y=−bax.

Want to see the full answer?

Chapter 7 Solutions

To Show:

Extended diagonals of the fundamental rectangle of the hyperbola x2a2−y2b2=1 are y=bax and y=−bax.