SOLUTION: Find the equation of the hyperbola with center at (4,-1) transverse axis parallel to the y axis, distance between foci is 10 and latus rectum is 9/2

The green line above, drawn between foci, is given to 
be 10 units long.  The center is its midpoint, so the 
two foci are (4,4) and (4,-6).
The two blue lines are the latus rectums. They are given
as 9/2, so by subtraction of half that or 9/4 from
the x-coordinate of the focus (4,4), we get that
the left end of the upper latus rectum is the point
(7/4,4).  The hyperbola goes through that point.

We know that the equation of the hyperbola is of the
form



and since the center is (4,-1), it's


or



Since it goes through (7/4,4), we substitute that 
for (x,y) 









We know that c = 5 because c is the distance from the
center to the focus.

For all hyperbolas,  or ,
and so 

Substitute in

















; 

;  

;  

b² can only be positive, so

b² = 9 

Substitute in 







So the equation:



becomes:


 
Edwin