The diagonals of a rectangle are both 10 and intersect at (0,0). Calculate the area of this rectangle, knowing that all of its vertices belong to the curve $y=\frac{12}{x}$.
At first I thought it would be easy - a rectanlge with vertices of (-a, b), (a, b), (-a, -b) and (a, -b). However, as I spotted no mention about the rectangle sides being perpendicular to the axises, it's obviously wrong which caused me to get stuck. I thought that maybe we could move in a similar way - we know that if a rectangle is somehow rotated (and we need to take that into account), the distances from the Y axis from the points being symmetric to (0,0) are still just two variables. So we would have: (-b, -12/b), (a, 12/a), (-a, -12/a), (b, 12/b). I then tried to calculate the distance between the first two and the second and the third which I could then use along with the Pythagorean theorem and a diagonal. However, the distance between the first two is $\sqrt{(a+b)^2+(\frac{12}{a}+\frac{12}{b})^2}$ which is unfriendly enough to make me thing it's a wrong way. Could you please help me?