Find an equation of the hyperbola having foci at 5 3 and 5 9

Find an equation of the hyperbola having foci at (-5, -3) and (-5, 9) and vertices at (-5, 1) and (-5, 5).

Solution

Given that the vertices and foci have common x-values, x=-5, that means the graph of this hyperbola has a vertical transverse axis.

The standard form of the equation of a hyperbola with a vertical transverse axis is as follows:

   (y - k)²/a² - (x - h)²/b² = 1

   where (h, k) is the center of the hyperbola, the vertices are at (h, k+a) and (h, k-a), and the foci are at (h, k+c) and (h, k-c).

Vertices: (-5, 1) = (h, k+a) ==> k + a = 1 ==> k = 1 - a

   (-5, 5) = (h, k-a) ==> k - a = 5 ==> k = 5 + a

   1 - a = 5 + a ==> 2a = -4 ==> a = -2 ==> a² = 4

Foci: (-5, -3) = (h, k+c) ==> k + c = -3 ==> k = -3 - c

   (-5, 9) = (h, k-c) ==> k - c = 9 ==> k = 9 + c

   -3 - c = 9 + c ==> 2c = -12 ==> c = -6 ==> c² = 36

Find b using the following formula: b² = c² - a²

   b² = 36 - 4 ==> b² = 32 ==> b = 42

Solve for k by plugging in appropriate variable into one of the equations determined for k:

   k = 1 - a ==> k = 1 - (-2) = 1 + 2 = 3 ==> k = 3

Thus, given that h = -5 , k = 3 , a² = 4 , and b² = 32 , the equation of the hyperbola is as follows:

   (y - 3)²/4 - (x + 5)²/32 = 1