1. Home
  2. Algebra and trigonometry
  3. Analytic geometry
  4. The hyperbola

12.2 The hyperbola  (Page 2/13)

<< Chapter < Page
  Algebra and trigonometry     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

Why is b in the answer
Dahsolar Reply
how do you work it out?
Brad Reply
answer
Ernest
heheheehe
Nitin
(Pcos∅+qsin∅)/(pcos∅-psin∅)
John Reply
how to do that?
Rosemary Reply
what is it about?
Amoah
how to answer the activity
Chabelita Reply
how to solve the activity
Chabelita
solve for X,,4^X-6(2^)-16=0
Alieu Reply
x4xminus 2
Lominate
sobhan Singh jina uniwarcity tignomatry ka long answers tile questions
harish Reply
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
ZAHRO Reply
If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .
Swetha Reply
Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?
Patrick Reply
Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?
Patrick
how I can simplify algebraic expressions
Katleho Reply
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
Mary Reply
23yrs
Yeboah
lairenea's age is 23yrs
ACKA
hy
Katleho
Ello everyone
Katleho
Laurene is 46 yrs and Mae is 23 is
Solomon
hey people
christopher
age does not matter
christopher
solve for X, 4^x-6(2*)-16=0
Alieu
prove`x^3-3x-2cosA=0 (-π<A<=π
Mayank Reply
create a lesson plan about this lesson
Rose Reply
Excusme but what are you wrot?
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, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask