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Finding the equation of a hyperbola centered at (0,0) given its foci and vertices

What is the standard form equation of the hyperbola    that has vertices ( ± 6 , 0 ) and foci ( ± 2 10 , 0 ) ?

The vertices and foci are on the x -axis. Thus, the equation for the hyperbola will have the form x 2 a 2 y 2 b 2 = 1.

The vertices are ( ± 6 , 0 ) , so a = 6 and a 2 = 36.

The foci are ( ± 2 10 , 0 ) , so c = 2 10 and c 2 = 40.

Solving for b 2 , we have

b 2 = c 2 a 2 b 2 = 40 36 Substitute for  c 2  and  a 2 . b 2 = 4 Subtract .

Finally, we substitute a 2 = 36 and b 2 = 4 into the standard form of the equation, x 2 a 2 y 2 b 2 = 1. The equation of the hyperbola is x 2 36 y 2 4 = 1 , as shown in [link] .

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What is the standard form equation of the hyperbola that has vertices ( 0, ± 2 ) and foci ( 0, ± 2 5 ) ?

y 2 4 x 2 16 = 1

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Hyperbolas not centered at the origin

Like the graphs for other equations, the graph of a hyperbola can be translated. If a hyperbola is translated h units horizontally and k units vertically, the center of the hyperbola    will be ( h , k ) . This translation results in the standard form of the equation we saw previously, with x replaced by ( x h ) and y replaced by ( y k ) .

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

The standard form of the equation of a hyperbola with center ( h , k ) and transverse axis parallel to the x -axis is

( x h ) 2 a 2 ( y k ) 2 b 2 = 1

where

  • the length of the transverse axis is 2 a
  • the coordinates of the vertices are ( h ± a , k )
  • the length of the conjugate axis is 2 b
  • the coordinates of the co-vertices are ( h , k ± b )
  • the distance between the foci is 2 c , where c 2 = a 2 + b 2
  • the coordinates of the foci are ( h ± c , k )

The asymptotes of the hyperbola coincide with the diagonals of the central rectangle. The length of the rectangle is 2 a and its width is 2 b . The slopes of the diagonals are ± b a , and each diagonal passes through the center ( h , k ) . Using the point-slope formula , it is simple to show that the equations of the asymptotes are y = ± b a ( x h ) + k . See [link] a

The standard form of the equation of a hyperbola with center ( h , k ) and transverse axis parallel to the y -axis is

( y k ) 2 a 2 ( x h ) 2 b 2 = 1

where

  • the length of the transverse axis is 2 a
  • the coordinates of the vertices are ( h , k ± a )
  • the length of the conjugate axis is 2 b
  • the coordinates of the co-vertices are ( h ± b , k )
  • the distance between the foci is 2 c , where c 2 = a 2 + b 2
  • the coordinates of the foci are ( h , k ± c )

Using the reasoning above, the equations of the asymptotes are y = ± a b ( x h ) + k . See [link] b .

(a) Horizontal hyperbola with center ( h , k ) (b) Vertical hyperbola with center ( h , k )

Like hyperbolas centered at the origin, hyperbolas centered at a point ( h , k ) have vertices, co-vertices, and foci that are related by the equation c 2 = a 2 + b 2 . We can use this relationship along with the midpoint and distance formulas to find the standard equation of a hyperbola when the vertices and foci are given.

Given the vertices and foci of a hyperbola centered at ( h , k ) , write its equation in standard form.

  1. Determine whether the transverse axis is parallel to the x - or y -axis.
    1. If the y -coordinates of the given vertices and foci are the same, then the transverse axis is parallel to the x -axis. Use the standard form ( x h ) 2 a 2 ( y k ) 2 b 2 = 1.
    2. If the x -coordinates of the given vertices and foci are the same, then the transverse axis is parallel to the y -axis. Use the standard form ( y k ) 2 a 2 ( x h ) 2 b 2 = 1.
  2. Identify the center of the hyperbola, ( h , k ) , using the midpoint formula and the given coordinates for the vertices.
  3. Find a 2 by solving for the length of the transverse axis, 2 a , which is the distance between the given vertices.
  4. Find c 2 using h and k found in Step 2 along with the given coordinates for the foci.
  5. Solve for b 2 using the equation b 2 = c 2 a 2 .
  6. Substitute the values for h , k , a 2 , and b 2 into the standard form of the equation determined in Step 1.

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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