If $A$ and $B$ are two points whose $(a{t}^2,2at),(\frac{a}{t^2},\frac{-2a}{t})$ co-ordinates are  and $S$ is a point $(a,0)$  then  $\frac{1}{SA}+\frac{1}{SB}=$

$A=(a{t}^2,2at),B=(\frac{a}{t^2},\frac{-2a}{t}),s(a,0)$

$SA=\sqrt{(a{t}^2-a^2)+{(0-2at)}^2}$, $SA=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

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

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

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

$=a(t^2+1)$

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

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

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

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

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

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

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

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

$=\frac{1}{a}$