i think for the 3rd one its better to go into parametric

let one end of focal chord be (at²,-2at)

since the end points of focal chord should satisfy the condi t₁.t₂=-1

by that the other end is(a/t²,2a/t).

let (h,k)be the point whose locus is to be foundi.e its the mid point of the above two points implies h=a(t²+1/t²)/2; k=2a(1/t-t)/2

now u can chek for the ans or else

calculate k² n (h-a)  n check which ans relates them

look this is not a shrt cut