Why did I get two solutions for B? (Rotation of Axes Problem)

[FabZeros]

I am having a very hard time with this problem... I tried to solve it by finding the equation of the ellipse in the rotated axis.

Using the angle measure obtained from using the inverse tan function, I tried to use theorems from the section to find the equation relative to the xy-axes.

I tried two different ways but became suspicious when I realized that either one or both of the values for B that I received are wrong...

How do I solve number 43?

 

[stapel]

Even if your images weren't upside-down, they're too faint for me to read. Please type out what you did, or else provide a much better (and right-side-up) graphic. Thank you!