1. Home
  2. College algebra
  3. Analytic geometry
  4. The hyperbola

8.2 The hyperbola  (Page 2/13)

<< Chapter < Page
  College algebra     Page 2 / 13
Chapter >> Page >

Deriving the equation of an ellipse centered at the origin

Let   ( − c , 0 )   and   ( c , 0 )   be the foci    of a hyperbola centered at the origin. The hyperbola is the set of all points   ( x , y )   such that the difference of the distances from   ( x , y )   to the foci is constant. See [link] .

If   ( a , 0 )   is a vertex of the hyperbola, the distance from   ( − c , 0 )   to   ( a , 0 )   is   a − ( − c ) = a + c .   The distance from   ( c , 0 )   to   ( a , 0 )   is   c − a .   The sum of the distances from the foci to the vertex is

( a + c ) − ( c − a ) = 2 a

If   ( x , y )   is a point on the hyperbola, we can define the following variables:

d 2 = the distance from  ( − c , 0 )  to  ( x , y ) d 1 = the distance from  ( c , 0 )  to  ( x , y )

By definition of a hyperbola,   d 2 − d 1   is constant for any point   ( x , y )   on the hyperbola. We know that the difference of these distances is   2 a   for the vertex   ( a , 0 ) .   It follows that   d 2 − d 1 = 2 a   for any point on the hyperbola. As with the derivation of the equation of an ellipse, we will begin by applying the distance formula    . The rest of the derivation is algebraic. Compare this derivation with the one from the previous section for ellipses.

                                       d 2 − d 1 = ( x − ( − c ) ) 2 + ( y − 0 ) 2 − ( x − c ) 2 + ( y − 0 ) 2 = 2 a Distance Formula ( x + c ) 2 + y 2 − ( x − c ) 2 + y 2 = 2 a Simplify expressions .                             ( x + c ) 2 + y 2 = 2 a + ( x − c ) 2 + y 2 Move radical to opposite side .                               ( x + c ) 2 + y 2 = ( 2 a + ( x − c ) 2 + y 2 ) 2 Square both sides .                      x 2 + 2 c x + c 2 + y 2 = 4 a 2 + 4 a ( x − c ) 2 + y 2 + ( x − c ) 2 + y 2 Expand the squares .                      x 2 + 2 c x + c 2 + y 2 = 4 a 2 + 4 a ( x − c ) 2 + y 2 + x 2 − 2 c x + c 2 + y 2 Expand remaining square .                                               2 c x = 4 a 2 + 4 a ( x − c ) 2 + y 2 − 2 c x Combine like terms .                                    4 c x − 4 a 2 = 4 a ( x − c ) 2 + y 2 Isolate the radical .                                        c x − a 2 = a ( x − c ) 2 + y 2 Divide by 4 .                                    ( c x − a 2 ) 2 = a 2 [ ( x − c ) 2 + y 2 ] 2 Square both sides .                      c 2 x 2 − 2 a 2 c x + a 4 = a 2 ( x 2 − 2 c x + c 2 + y 2 ) Expand the squares .                     c 2 x 2 − 2 a 2 c x + a 4 = a 2 x 2 − 2 a 2 c x + a 2 c 2 + a 2 y 2 Distribute  a 2 .                                    a 4 + c 2 x 2 = a 2 x 2 + a 2 c 2 + a 2 y 2 Combine like terms .                   c 2 x 2 − a 2 x 2 − a 2 y 2 = a 2 c 2 − a 4 Rearrange terms .                     x 2 ( c 2 − a 2 ) − a 2 y 2 = a 2 ( c 2 − a 2 ) Factor common terms .                               x 2 b 2 − a 2 y 2 = a 2 b 2 Set  b 2 = c 2 − a 2 .                              x 2 b 2 a 2 b 2 − a 2 y 2 a 2 b 2 = a 2 b 2 a 2 b 2 Divide both sides by  a 2 b 2                                      x 2 a 2 − y 2 b 2 = 1

This equation defines a hyperbola centered at the origin with vertices   ( ± a , 0 )   and co-vertices   ( 0 ± b ) .

Standard forms of the equation of a hyperbola with center (0,0)

The standard form of the equation of a hyperbola with center   ( 0 , 0 )   and transverse axis on the x -axis is

x 2 a 2 − y 2 b 2 = 1

where

  • the length of the transverse axis is   2 a
  • the coordinates of the vertices are   ( ± a , 0 )
  • the length of the conjugate axis is   2 b
  • the coordinates of the co-vertices are   ( 0, ± b )
  • the distance between the foci is   2 c , where   c 2 = a 2 + b 2
  • the coordinates of the foci are   ( ± c , 0 )
  • the equations of the asymptotes are   y = ± b a x

See [link] a .

The standard form of the equation of a hyperbola with center   ( 0 , 0 )   and transverse axis on the y -axis is

y 2 a 2 − x 2 b 2 = 1

where

  • the length of the transverse axis is   2 a
  • the coordinates of the vertices are   ( 0, ± a )
  • the length of the conjugate axis is   2 b
  • the coordinates of the co-vertices are   ( ± b , 0 )
  • the distance between the foci is   2 c , where   c 2 = a 2 + b 2
  • the coordinates of the foci are   ( 0, ± c )
  • the equations of the asymptotes are   y = ± a b x

See [link] b .

Note that the vertices, co-vertices, and foci are related by the equation   c 2 = a 2 + b 2 .   When we are given the equation of a hyperbola, we can use this relationship to identify its vertices and foci.

Questions & Answers

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
John Reply
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Ummi Reply
omeprazole
Kamara
what
Renee
what is this
Renee
is a drug
Kamara
of anti-ulcer
Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
Nency Reply
Not really sure
Eli
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
what is anatomy
Oyindarmola Reply
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Oyerinde Reply
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Mohammed Reply
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
Wulku Reply
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
Ali Reply
function of digestive
Ali
the diagram of the lungs
Adaeze Reply
what is the normal body temperature
Diya Reply
37 degrees selcius
Xolo
37°c
Stephanie
please why 37 degree selcius normal temperature
Mark
36.5
Simon
37°c
Iyogho
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
37A c
Wulku
what is anaemia
Diya Reply
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
Diya Reply
how does Lysin attack pathogens
Diya
acid
Mary
I information on anatomy position and digestive system and there enzyme
Elisha Reply
anatomy of the female external genitalia
Muhammad Reply
Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)
Theophilus Reply
what's lochia albra
Kizito
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 4

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask