Question

Write the equation of the hyperbola.

Foci: (-9,0), (9,0)
Vertices: (-4,0), (4,0)

Answer 1

Hyperbola is a line in which the distance from each focus subtracted gets some constant.
\[
\sqrt{(x-9)^2+(y-0)^2} - \sqrt{(x-(-9))^2+(y-0)^2} = c
\]
We can can find \(c\) by looking at the focus and vertices, since the vertices are on the hyperbola.
\[
c = ||-9-4| - |9-4|| = |15 - 5| = 10
\]

Answer 2

by line, I meant curve

Answer 3

There is some other equation for it too... something like: \[
\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1
\]But that requires you to know what \(a\) and \(b\) are.
I think \(a\) is like the distance of the vertices and \(a/b\) is the slope of the locus... but I am not completely sure.

Answer 4

@wio thank you so much, what would the final equation be in that case?

Answer 5

I dunno, I'd have to really try hard at it.