Question

Let F = (0,9) be the focus and the line y = 1 be the directrix. Plot several points P that are three times as far from the focus as they are from the directrix, including the vertices on the y-axis. The configuration of all such P is a hyperbola of eccentricity 3. Use the distance formula to write an equation for the hyperbola. Find the values of a, b, and c for this curve, then calculate the ratio c/a. Is the result what you expected?

Answer 1

since e = 3, c/a = 3
Now, review the properties of an hyperbola.
Given the center at (0,0), with vertex at (0,a), focus at (0,c) and directrix at y=d, we have
(c-a)/(a-d) = 3
c/a = 3
c-d = 8
Solve that for a,c,d, and you have
a=3, c=9, d=1
Since we know the directrix is at y=1, the center must be at y = 1
b^2 = c^2-a^2 = 72

So, our hyperbola is

y^2/9 - x^2/72 = 1
See its properties at
https://www.wolframalpha.com/input/?i=hyperbola+y%5E2%2F9+-+x%5E2%2F72+%3D+1

It does not list the directrices, but we know that
d = a^2/c = 9/9 = 1