The point of the intersection of the two tangents to the ellipse 2x² + 3y² = 6 at the end of latus rectum is

$\mathrm{e}=\text{ eccentricity }=\sqrt{1-{\left(\frac{\mathrm{b}}{\mathrm{a}}\right)}^2}$$\Rightarrow \mathrm{e}=\sqrt{\frac{1}{3}}$

Ends of latus rectum of the ellipse are $\left(ae,\pm \frac{b^2}{a}\right)=\left(1,\pm 2/\sqrt{3}\right)$

Equation of tangents are

$\begin{array}{r}2x_1+3{\mathrm{yy}}_1=6\Rightarrow 2\mathrm{x}+2\sqrt{3}\mathrm{y}=6\\ 2{\mathrm{xx}}_2+3{\mathrm{yy}}_2=6\Rightarrow 2\mathrm{x}-2\sqrt{3}\mathrm{y}=6\end{array}$

Point of intersection of tangents at the ends of latus rectum is

$\mathrm{z}=\left(\frac{\mathrm{a}}{\mathrm{e}},0\right)$=(3,0)