If $y = mx + c$ is a tangent to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$,then the value of $c$ is ...

The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse $x^2+9y^2=9$ meets its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A$,$M$,and the origin $O$ is

If the line $x - 2y = 12$ is tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at the point $(3, -4.5)$,then the length of the latus rectum of the ellipse is:

For the ellipse given by $\frac{(x-3)^2}{25}+\frac{(y-2)^2}{16}=1$,match the equations of the lines given in List-$I$ with those on the List-$II$.

List-$I$	List-$II$
$(i)$ The equation of the major axis	$(p)$ $3x = 34$
$(ii)$ The equation of a directrix	$(q)$ $y = 2$
$(iii)$ The equation of a latus rectum	$(r)$ $x + y = 9$
	$(s)$ $x = 6$
	$(t)$ $x = 3$
	$(u)$ $3y = 34$

Statement $I$: The equation of the directrix of the ellipse $4x^2+y^2-8x-4y+4=0$ is $3y=6-4\sqrt{3}$.
Statement $II$: The equation of the latus rectum of the ellipse $x^2+4y^2-4x-8y+4=0$ is $y=2+\sqrt{3}$.
Which of the above statement$(s)$ is (are) true?

If the distance between the directrices is thrice the distance between the foci,then the eccentricity of the ellipse is