Intermediate Problems (2)

Two perpendicular chords of a circle $A_{1}A_{4}$ and $A_{2}A_{3}$ intersect at a point $P$ . If for $i=1, 2, 3$ , the length of $PA_{i}=2^{i}$ , the radius of the circle can be expressed as $\sqrt{a}$ where $a$ is a positive integer, find $a$ .

This problem is part of the set Intermediate Problems