The length of the latusrectum of an ellipse is 6 units and the distance between a focus and its nearest vertex on the major-axis is $5 / 3$ units. If $e$ is the eccentricity of this ellipse, then $e$ satisfies the equation

If the line $2 x-3 y+4=0$ cuts the ellipse $x=3 \cos \theta, y=5 \sin \theta$ in $A$ and $B$ and $(\alpha, \beta)$ is the mid-point of $A B$, then $3 \beta-2 \alpha=$

Statement I The equation of the directrix of the ellipse $4 x^2+y^2-8 x-4 y+4=0$ is $3 y=6-4 \sqrt{3}$

Statement II The equation of the latusrectum of the ellipse $x^2+4 y^2-4 x-8 y+4=0$ is $y=2+\sqrt{3}$

Which of the above statement(s) is (are) true?

If $S$ is the focus of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ lying on the positive $X$ - axis and $P(\theta)$ is a point on the ellipse such that $S P=1$, then $\cos \theta=$