We find a² and b² the same way we find them if the center is the origin. However we must also find the center of the hyperbola. We will do this by finding the midpoint of the segment that joins the two foci. Lets explore what we mean.

For this example, lets say that we are given the two foci are (2,-3) and (2,7) and the difference of the focal radii is 8. We find a², b² and c² like we have before. 2c = 10 because the distance between the two foci is found by subtracting the two coordinates that are different. This makes c = 5 and c² = 25. Using the fact that the difference of the focal radii is equal to the major axis, we get 2a = 8 making a = 4 and a² = 16. Then we substitute that information in the equation b² = c² - a² giving us b² =25 - 16, b² =9.

Now we find the center by finding the midpoint of the foci like we said earlier. We add the x coordinates and divide by 2...which is ( 2+ 2)/2 and do the same with the y coordinates...(-3 + 7)/2. This gives us the center of (2, 2). Subtract these two coordinates from the x and y variable when we write our final equation.

We also need to determine which is the major axis by connecting the two foci to see if a vertical line is formed making y the major axis or if a horizontal line is formed, making x the major axis. Remember the major axis determines which variable comes before the subtraction sign. In this case they form a vertical line, so y is the major axis.

We finally have all the information to write our final equation...take the values of a², b² and the center and try to write the correct equation. Then check it below.

Since y is the major axis, it comes before the subtraction sign and a² goes under it, then x ² goes after the subtraction sign with b² under it, keeping in mind you must subtract the center also...So your final equation should be:

(y - 2)²     (x - 2)²
----------- - ------------   =  1
  16             9