This question corresponds to an ellipse , the angle of triangle are $60^°,30^° ,90^°$. I tried to use the formula that product of distance drawn perpendicular to major axis from focus to the tangent is equal to $b^2$ if $a>b$ and ellipse is $*\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, but not able to proceed
Hints only
The ellipse with maximum area is Steiner inellipse.
Use Marden's Theorem to find out the foci.
Let the cubic be $$z(z-1)(z-i\sqrt{3})=0$$
See also another post of mine for your interests.
Further points
Family of inscribed ellipse bounded by the axes and line $\dfrac{x}{a}+\dfrac{y}{b}=1$ is given by $$ \left( \frac{x}{\lambda a}+\frac{y}{\mu b}-1 \right)^2=\frac{4xy}{ab} \left( \frac{1}{\lambda}-1 \right) \left( \frac{1}{\mu}-1 \right) $$ where $\lambda,\mu \in (0,1)$.
Area of the ellipse is $$\frac{\pi \lambda \mu ab}{2} \sqrt{\frac{(1-\lambda)(1-\mu)}{(\lambda+\mu-\lambda \mu)^3}}$$
