<Cir-2024-767>, hint

Let PA=PB=r (radius), <PMA=x. MP=p

PA^2=r^2+p^2-2pr cosx,   PB^2=r^2+p^2-2pr cos(180-x)

PA^2+PB^2=2(r^2+p^2)=constant

Let center of  sphere ,its  radius be O, R and suppose line MO cuts sphere at T and T'

be the other point on S. Then  in triangle MOT' OT=OT',OT'+T'M>OM=OTM. T'M>TM

Hence AT^2+BT^2 is minimum