Scaling down a set of points into a smaller area
The following procedure achieves your requirements:
Step 1: Find the smallest circle that encloses all points.  (There are standard algorithms for this.)
Step 2: Compute the area of that circle.
Step 3: Compute the scaling ratio $c$ needed to achieve the desired area.  (For instance if the area of that circle is 100 and you want area 5, then your scaling ratio is $c = \sqrt{100/5} = \sqrt{20}$.)
Step 4: Scale everything down by a factor of $c$, centered at the center of the circle found in Step 1.
This can be implemented in linear time, so it should be very efficient.