A hyperbola is the set of all points around two foci, or focus points, such that the difference of the distances from any point to each focus is a positive constant.
The standard form of the equation for this type of hyperbola given the center is at the origin is:
(y^2 / a^2) - (x^2 / b^2) =1
Your coordinates of the vertices are (0, +/-9) and (0 +/-11) making a=9 and c=11.
To find the equation of the vertical hyperbola, we'll first need to use the 'a' and 'c' values to find 'b' using the formula b^2=c^2 - a^2.
Remember, 'a' is the radius of the major or transverse axis, 'b' is the radius of the minor or conjugate axis, and 'c' is the distance between the focus and the center. If we work our formula to solve for 'b' and plug in the values we get:
b^2 = 11^2 - 9^2
b^2 = 40
Plug in the values from above b^2 = 105 and a^2 = 16
(y^2/81) - (x^2/40) = 1
The least common multiple of the denominator is 3240, so multiply each term by 3240 to get rid of the fractions. This result is the equation for this hyperbola.
40y^2 - 81x^2 = 3240
