Question

Find an equation in standard form for the hyperbola with vertices at (0, ±2) and foci at (0, ±11). @owlcoffee

Answer 1

Do you know how to do this @Owlcoffee

Answer 2

The hyperbola is defined as: "The geometric place composed of all the points in the plane such that the absolute value of the difference of their distance to two fixed points called "foci" is equal to a given measure smaller than the distance between the foci".

Little confusing, but in a mathematical term it woul look like this:
\[\left| PF -PF' \right|=2a\]
Where "2a" is but a constant where "a" is a real number.

But, unlike when we were dealing with a line or a circumference, a hyperbola does not give away the components easily.
The given focis usually give away what type of hyperbola which in this case are F1(0,11) and F2(0,-11), this means that the focal axis is inside the "y-axis", since those are the points that satisfy x=0.
A hyperbola with that focal axis has the equation of:
\[\frac{ y^2 }{ a^2 }-\frac{ x^2 }{ b^2 }=1\]
"a" is the distance between the vertices and "b" is a component of the "conjugate axis" and we will call "c" the focal distance.
The relationship between these points is:
\[c^2=a^2+b^2\]