In working through a theory of mine, I have come across a curve I cannot identify. Along with x and y, this curve needs an additional input to complete the curve. Things I know about this curve:

The curve is identical when reflected around the line y=x.

The curve always contains the points (0,10), (10,0), (5,a), and (a,5).

The area under the curve from 0 to 10 is always 10a.

At a=5, the curve matches perfectly with the line 10-x.

I am nearly certain that the curve is hyperbolical in nature.

Knowing this, what is the equation of this curve?