From pythagreon theorem, you know that
20=sqrt(a^2+b^2) -> 400=a^2 + b^2 (Eq. 1)
You also know that the area of the triangle will be
A=ab/2 (Eq. 2)
Now solve the first equation for b, and sub the result into Eq. 2. Then differentiate wrt a. Set the derivative equal to zero and solve for a. You will get two values of a, one of which is inadmissable since you are trying to maximize the area. When you have found the correct value of a, sub it into Eq. 1 to solve for b.
Hope this helps!