PQ2=S1=h2+k2−a2....(1)
Any secant through P(h,k) is x−hcosθ=y−ksinθ=r
where r is the distance of P from any point on the line.
(rcosθ+h,rsinθ+k) is any point on the line.
If it lies on the circle, then
(rcosθ+h)2+(rsinθ+k)2=a2
or r2+2r(hcosθ+ksinθ)+(h2+k2−a2)=0
It gives two values of r say PA and PB.
PA.PB=r1r2=h2+k2−a2=S2
=PQ2=constant, by (1)
as it is independent of θ. Hence all the products are equal and each equals to PQ2.