there are 3 (one-parameter) families which passes through the vertices of this square:
2. hyperbolae opening to the right and the left
3. hyperbolae opening up and down
The center of all conics is the origin.
to #1.: The general equation of an ellipse with center at (0, 0) and the axes a and b is:
..........Plug in the coordinates of (-1, 1) and (1, 1):
.......... Plug in this term for bē into the original equation of the ellipse:
Don't forget to calculate the domain of the family!
To demonstrate the effect of this equation I've attached a drawing with 1 < a < 5
To #2, #3: Use the same method.