How do you identify the vertices, foci, and direction of #x^2/81-y^2/4=1#?

1 Answer
Jan 2, 2017

Please see the explanation.

Explanation:

The reference Conics: Hyperbolas tells us that the given equation:

#x^2/81 - y^2/4 = 1" [1]"#

is that of a hyperbola with a horizontal transverse axis. We can identify that the direction is horizontal, because the "x" term is positive and the "y" term is negative.

NOTE: If the "y" term were positive and "x" term were negative, then the direction would be vertical.

The reference, also, tells us that the following equation, [2], is the standard Cartesian form:

#(x - h)^2/a^2 - (y - k)^2/b^2 = 1" [2]"#

In this form, it is easy to observe or compute:

  1. The center is #(h, k)#
  2. The vertices are located at #(h - a, k) and (h + a, k)#
  3. The foci are located at #(h - sqrt(a^2 + b^2), k) and (h + sqrt(a^2 + b^2), k)#

Write equation [1] in the form of equation [2]:

#(x - 0)^2/9^2 - (y - 0)^2/2^2 = 1" [3]"#

In this form, the vertices can be written by observation:

#(-9, 0)# and #(9,0)#

Compute #sqrt(a^2 - b^2) = sqrt(9^2 + 2^2) = sqrt(85)#

The foci are:

#(-sqrt(85), 0)# and #(sqrt(85),0)#