Simple Parametric question... (1 Viewer)

[jassneetm]

using P (2ap,ap^2), Q (2aq,aq^2), S (0,a)
Show that 1/QS+1/PS=1/a where QS and PS are distances.

 

[Riviet]

For this to work, we need to assume that the focus, S is a focal chord. By substituting the focus (0,a) into the equation of the chord PQ, we find that pq=-1.

LHS=1/QS + 1/PS

=1/sqrt{(2aq)²+(aq²-a)²)} + 1/sqrt{(2ap)²+(ap²-a)²}

=1/sqrt{(aq²+a)²} + 1/sqrt{(ap²+a)²)}, by expanding numerator, collecting like terms and re-factorising

= 1/a.{1/(q²+1) + 1/(p²+1)}

=1/a.{(p²+q²+2)/[(p²+1)(q²+1)]}

=1/a.{(p²+q²+2)/(p²+q²+1+(pq)²)}

=1/a.{(p²+q²+2)/(p²+q²+2)}, using pq=-1

=1/a

=RHS