Question

if P and Q are two points whose coordinates are (at^2,2at) and (a/t^2,-2a/t) respectively and S (a,0) , show that 1/SP + 1/SQ is independent of t.

Answer 1

This seems to be related to a parabola   y² = 4 a x

P(at², 2at)     Q(a/t²,  -2a/t)      S(a, 0)

SP² = 4a²t² + a²(t⁴- 2t² + 1) = a²(t²+1)²
           SP = a(t²+1)
SQ² = 4a²/t² + a² (1/t⁴ + 1 - 2/t²) = a² (1/t²  +1)²
           SQ = a (t²+1) /t²

1/SP + 1/SQ = (1 + t²) /[ a (t²+1) ] =   1/a

Since the sum is independent of  the parameter t,  the sum is constant for all point pairs P & Q.