The angle between the pair of tangents drawn from the point $(1, 2)$ to the ellipse $3{x^2} + 2{y^2} = 5$ is

If the tangent at a point on the ellipse $\frac{{{x^2}}}{{27}} + \frac{{{y^2}}}{3} = 1$ meets the coordinate axes at $A$ and $B,$  and  $O$  is the origin, then the minimum area (in sq. units) of the triangle $OAB$  is

If the area of the auxiliary circle of the ellipse $\frac{{{x^2}}}{{{a^2}}}\, + \,\frac{{{y^2}}}{{{b^2}}}\, = \,1(a\, > \,b)$  is twice the area of the ellipse, then the eccentricity of the  ellipse is

The equation of an ellipse, whose vertices are $(2, -2), (2, 4)$ and eccentricity $\frac{1}{3}$, is

If the length of the latus rectum of the ellipse $x^{2}+$ $4 y^{2}+2 x+8 y-\lambda=0$ is $4$ , and $l$ is the length of its major axis, then $\lambda+l$ is equal to$......$

If $C$ is the centre of the ellipse $9x^2 + 16y^2$ = $144$ and $S$ is one focus. The ratio of $CS$ to major axis, is