If  $R(2,6),S(12,K)$ are the foci of an ellipse which touches both the coordinate axes then

$x=0,y=0$  are tangents
$\therefore b^2=2.12=6.K\Rightarrow b^2=24,K=4$ ( product of perpendicular distances from foci to tangents)
$2ae=$  distance between foci  $=\sqrt{{10}^2+2^2}=\sqrt{104}$
 $4(a^2-b^2)=104\Rightarrow a^2=50$
 $\therefore e=\sqrt{\frac{50-24}{50}}=\frac{\sqrt{13}}{5}$
LLR  $=\frac{2b^2}{a}=\frac{2.24}{5\sqrt{2}}=\frac{24\sqrt{2}}{5}$
Length of major axis  $=2a=10\sqrt{2}$
Double ordinate is minor axis
  Length  $=2b=4\sqrt{6}$