Straight Lines and Pair of Straight Lines

Let S denote the locus of the point of intersection of the pair of lines
‌4x−3y=12α
‌4αx+3αy=12
where α varies over the set of non-zero real numbers. Let T be the tangent to S passing through the points (p,0) and (0,q),q>0, and parallel to the line 4x−‌

3

√2

y=0.
Then the value of pq is