Answer : $6+2\sqrt 5,2\sqrt 5$
It is given that the axes are 6 cm and 4 cm.
Hence $a=6\Rightarrow a^2=36$
$b=4\Rightarrow b^2=16$
We know $b^2=a^2(1-e^2)$
(ie) $16=36(1-e^2)$
$\therefore e^2=1-\large\frac{4}{9}=\frac{5}{9}$
$e=\large\frac{\sqrt 5}{3}$
$\therefore ae=\large\frac{6\times \sqrt 5}{3}$
$\Rightarrow 2 \sqrt 5$
Hence the length of the string and distance between the pins are $6+2\sqrt 5,2\sqrt 5$