Explanation Hyperbola is a curve formed by tracing the locus point P in a plane surface that satisfy a specific condition with two fixed points named as F and F ' . The specific condition on the points P , F and F ' for constructing a hyperbola curve is as follows: | d ( P F ) − d ( P F ' ) | = constant Here, d ( P F ) is the distance between points P and F . Similarly d ( P F ' ) is the distance between points P and F ' . Where P describe all the points that trace a hyperbola, points F and F ' are fixed points. There are two standard forms of the Hyperbola obtained as shown below: The standard form of an hyperbola with center at ( h , k ) and transverse axis horizontal is ( x − h ) 2 a 2 − ( y − k ) 2 b 2 = 1 . The standard form of an hyperbola with center at ( h , k ) and transverse axis vertical is ( y − k ) 2 a 2 − ( x − h ) 2 b 2 = 1 . Both standard equations are similar to the form ( x − m ) 2 n 2 − ( y − p ) 2 q 2 = 1 . Consider the first standard form of an hyperbola ( x − h ) 2 a 2 − ( y − k ) 2 b 2 = 1 and expand it. ( x − h ) 2 a 2 − ( y − k ) 2 b 2 = 1 b 2 ( x − h ) 2 − a 2 ( y − k ) 2 = a 2 b 2 b 2 ( x 2 + h 2 − 2 h x ) − a 2 ( y 2 + k 2 − 2 k y ) = a 2 b 2 b 2 x 2 + b 2 h 2 − 2 b 2 h x − a 2 y 2 − a 2 k 2 + 2 a 2 k y = a 2 b 2 Now, arrange in descending order of power of the variables as follows: b 2 x 2 + b 2 h 2 − 2 b 2 h x − a 2 y 2 − a 2 k 2 + 2 a 2 k y = a 2 b 2 b 2 x 2 − a 2 y 2 − 2 b 2 h x + 2 a 2 k y + b 2 h 2 − a 2 k 2 − a 2 b 2 = 0 ( b 2 ) x 2 + ( 0 ) x y + ( − a 2 ) y 2 + ( − 2 b 2 h ) x + ( 2 a 2 k ) y + ( b 2 h 2 − a 2 k 2 − a 2 b 2 ) = 0 As a , b , h and k are constants, the above equation is in the form of the A x 2 + B x y + C y 2 + D x + E y + F = 0 with B = 0

WebAssign with Corequisite Support for Gustafson/Hughes' College Algebra, 12th Edition [Instant Access], Single-Term

12th Edition

ISBN: 9780357422533

Author: Gustafson; R. David; Hughes; Jeff

Publisher: Cengage Learning US

See similar textbooks

Videos

Question

Chapter 7.3, Problem 71E

To determine

To Show:

The expansion of the standard equation of a hyperbola is a special case of the general second-degree equation in two variables with B=0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.3, Problem 70E

Chapter 7.3, Problem 72E

Students have asked these similar questions

(1) Let M and N be non-empty subsets of a linear space X, show that whether = U or not, and show that there whether exsits a liear function from P₂(x) into R' which onto but not one-to-one or not. ام (2) Let R be a field of real numbers and P,(x)=(a+bx+cx? / a,b,ce R} be a vector space over R, show that whether there exsit two hyperspaces A and B such that AUB is a hyperspace or not. (3) Let A be an affine set in a linear space X over afield F and tEA, show that A-t is a subspace of Xand show that if M and N are balanced sets then M+N is balanced set. (4) Write the definition of bounded set in a normed space, and write with prove an equivalent statement to definition. (5) Let d be a metric on a linear space X over a field F, write conditions on d in order to get that there is a norm on X induced dy d and prove that. (6) Let M be a non-empty subset of a normed space X, show that xEcl(M) iff for any r>o there exsits yEM such that llx-yll

Find all solutions to the following equation. Do you get any extraneous solutions? Explain why or why not. 2 2 + x+1x-1 x21 Show all steps in your process. Be sure to state your claim, provide your evidence, and provide your reasoning before submitting.

Directions: For problems 1 through 3, read each question carefully and be sure to show all work. 1. What is the phase shift for y = 2sin(2x-)? 2. What is the amplitude of y = 7cos(2x+л)? 3. What is the period of y = sin(3x-π)? Directions: For problems 4 and 5, you were to compare and contrast the two functions in each problem situation. Be sure to include a discussion of similarities and differences for the periods, amplitudes, y-minimums, y-maximums, and any phase shift between the two graphs. Write in complete sentences. 4. y 3sin(2x) and y = 3cos(2x) 5. y 4sin(2x) and y = cos(3x- -플)

Chapter 7 Solutions

WebAssign with Corequisite Support for Gustafson/Hughes' College Algebra, 12th Edition [Instant Access], Single-Term

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.

The expansion of the standard equation of a hyperbola is a special case of the general second-degree equation in two variables with B = 0 .

Solution Summary: The author explains that hyperbola is a curve formed by tracing the locus point P in the plane surface.

Videos

To Show:

The expansion of the standard equation of a hyperbola is a special case of the general second-degree equation in two variables with B=0.

Want to see the full answer?

Chapter 7 Solutions

The expansion of the standard equation of a hyperbola is a special case of the general second-degree equation in two variables with B = 0 .

Solution Summary: The author explains that hyperbola is a curve formed by tracing the locus point P in the plane surface.

Videos

To Show: The expansion of the standard equation of a hyperbola is a special case of the general second-degree equation in two variables with B=0.

Want to see the full answer?

Chapter 7 Solutions

To Show:

The expansion of the standard equation of a hyperbola is a special case of the general second-degree equation in two variables with B=0.