Question

If P and Q are two points whose coordinates are $(a{t}^2,2at)$ and $(\frac{a}{t^2},\frac{2a}{t})$ respectively and S is the point $(a,0)$. Show that $\frac{1}{SP}+\frac{1}{SQ}$ is independent of t.

Answer 1

$SP=\sqrt{(a{t}^2-a{)}^2+(2at-0{)}^2}$

$SP=a(t^2+1)$

And,

$SQ=\sqrt{(\frac{a}{t^2}-a{)}^2+(\frac{2a}{t}-0{)}^2}$

$SQ=\sqrt{\frac{a^2(1-t^2{)}^2}{t^4}+\frac{4a^2}{t^2}}$

$SQ=\frac{a}{t^2}(1+t^2)$

Therefore,

$\frac{1}{SP}+\frac{1}{SQ}=\frac{1}{a(t^2+1)}+\frac{t^2}{a(t^2+1)}=\frac{1}{a}$, which is independent of t.