There is one more method to do this question.

that is by using the equation of the chord whose mid pt. is given  S₁=T

let the mid pt be (a,0)

Therefore we get     a²-ap=ax -px/2 -ap/2 - qy/2

now since pt. (p,q) lies on chord we can satisfy the equation with it

a²-ap-ap+p²/2+ap/2+q²/2=0

2a²-4ap+p²+ap+q²=0

2a²-3ap+(p²+q²)=0

Now,this quadratic shud have two distinct roots as two distinct are bisected by x-axis,hance the discriminant D>0

9p²-8(p²+q²)>0

thus we get p²>8q²

Plz rate if the soln. was helpful.