Question #40787

Let $T$ be a triangle in the $xy$-plane with an area of $50$ sq. units. A new coordinate system $(X, Y)$ is defined by shifting the origin and applying a linear transformation such that the new coordinates are given by: $X = (e^\alpha - e^{-\alpha})^2 x + (e^{2\alpha} - e^{-2\alpha}) y - \ln 2$ $Y = (e^{2\alpha} - e^{-2\alpha}) x + (e^\alpha + e^{-\alpha})^2 y + \ln 3$ for some real constant $\alpha \neq 0$. What is the area of the triangle $T$ when measured in the $XY$-coordinate system?