Foci and the definition of a hyperbola

[darshanpatel]

Homework Statement

Find an equation of a hyperbola such that for any point (x,y) on the hyperbola, the difference between its distances from the points (2,2) and (10,2) is 6.

Homework Equations

-None-

The Attempt at a Solution

I tried graphing it and making the (10,2) and (2,2) the vertices of the graphs. They open left to right because the points are horizontal to each other.

Equation for that type of circle: (y-k)^2/(a^2)-(x-h)^2/(b^2)=1

Distance between the two vertices =6 so center is (3,2)

I don't know what to do to get the a and b values because the equation right now would be:

(((y-2)^2)/(a^2))-(((x-3)^2)/(b^2))=1

Also is that right this far?

 

[scurty]

I'll have to think a bit about the answer, but I can tell you right now that the answer isn't what you are describing. The vertices you are describing are 8 units apart, not 6 (10-2=8). Also, the midpoint of the x values of the line segment connecting the two points would be \frac{10+2}{2} = 6, so the midpoint would be (6,2), not (3,2).

Edit: Okay, I think I have it now. What are the two points in the plane called where, for every point on the hyperbola, the absolute value of the difference of the distance to each of those two points is a constant? How does that constant relate to the equation of the hyperbola?