Perpendicular distance of directrix from focus
$$\eqalign{ & = \frac{a}{e} - ae = 4 \cr & \Rightarrow a\left( {2 - \frac{1}{2}} \right) = 4 \cr & \Rightarrow a = \frac{8}{3} \cr} $$
$$\therefore $$ Semi major axis $$ = \frac{8}{3}$$

The equation of the pair of tangents is $$S{S_1} = {T^2}$$
or $$\left( {\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} - 1} \right)\left( {\frac{{{\lambda ^2}}}{9} + \frac{9}{4} - 1} \right) = {\left( {\frac{{\lambda x}}{9} + \frac{{3y}}{4} - 1} \right)^2}$$
For right angle, $$a + b = 0 \Rightarrow \left\{ {\frac{1}{9}\left( {\frac{{{\lambda ^2}}}{9} + \frac{5}{4}} \right) - \frac{{{\lambda ^2}}}{{81}}} \right\} + \left\{ {\frac{1}{4}\left( {\frac{{{\lambda ^2}}}{9} + \frac{5}{4}} \right) - \frac{9}{{16}}} \right\} = 0$$