JEE Coordinate (3)!

Variable pairs of chords at right angles and drawn through a point $P$ (with eccentric angle $\frac\pi4$ ) on the ellipse $\dfrac{x^2}4+y^2 = 1$ to meet the ellipse at two points, say $A$ and $B$ .

If the line joining $A$ and $B$ passes through a fixed point $Q=(a,b)$ and the value of $a^2+b^2$ can be expressed as $\dfrac mn$ , where $m$ and $n$ are coprime positive integers, submit your answer as $n-m$ .

This is a part of my set Practice for JEE 2017!