P and Q are the points with parameters p and q on the parabola $\displaystyle x=2at , y=at^2$
If pq = 1 and S is the focus of the parabola, show that
1/SP + 1/SQ = 1/a
Thanks
The focus is at $\displaystyle S = (0,a)$. If $\displaystyle P = (2ap,ap^2)$ and $\displaystyle Q = (2aq,aq^2)$ then $\displaystyle SP^2 = 4a^2p^2 + a^2(1-p^2)^2 = a^2(1+p^2)^2$ and so $\displaystyle SP = a(1+p^2)$. Similarly $\displaystyle SQ = a(1+q^2)$.
Then $\displaystyle \frac1{SP} + \frac1{SQ} = \frac1{a(1+p^2)} + \frac1{a(1+q^2)} = \frac{2+p^2+q^2}{a(1+p^2)(1+q^2)}$. If $\displaystyle pq=1$ then you can write the numerator as $\displaystyle 1+p^2+q^2+p^2q^2 = (1+p^2)(1+q^2)$, so the fraction simplifies to $\displaystyle \frac1a$.