Let a circle be given by $2x(x-a)+y(2y-b)=0;$ $(a\ne 0, b\ne 0)$. Find the condition on $a$ and $b$ if two chords, each bisected by the $x$-axis, can be drawn to the circle from the point $\left(a,\cfrac{b}{2}\right)$.

The circle equation is 2x^2 - 2ax + 2y^2 - by = 0. The center is (a/2, b/4). A chord bisected by the x-axis means the midpoint of the chord has y=0. For two such chords to exist from (a, b/2), the point must lie outside the circle, and the geometry of the circle must allow for the intersection of the chord with the x-axis.