Correct answer is: 61
\(\frac{2 \mathrm{~b}^2}{\mathrm{a}}=9 \text { and } \pm \frac{\mathrm{a}}{\mathrm{e}}= \pm \frac{4}{\sqrt{3}}\) ....(1)
Equation of tangent \(y=\sqrt{3} x-\sqrt{3}\)
m = \(\sqrt3\)
c = \(-\sqrt3\)
Condition of tangency \(c= \pm \sqrt{a^2 m^2-b^2}\)
\(-\sqrt{3}=\sqrt{3 a^2-b^2}\)
\( 3=3 a^2-b^2\)
\( 6 a^2-9 a-6=0\)
\( 2 a^2-3 a-2=0\)
\( 2 a^2-4 a+a-2=0\)
\( (a-2)(2 a+1)=0\)
\( a=2 \text { or }-\frac{1}{2}\)
when \(\mathrm{a}=2, \mathrm{~b}^2=9\) put a = 2 in ...(1)
\( \frac{2}{\mathrm{e}}=\frac{4}{\sqrt{3}} \Rightarrow \mathrm{e}=\frac{\sqrt{3}}{2}\)
Which is impossible
