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 position of the point $(1, 3)$ with respect to the ellipse $4{x^2} + 9{y^2} - 16x - 54y + 61 = 0$

Consider two straight lines, each of which is tangent to both the circle $x ^2+ y ^2=\frac{1}{2}$ and the parabola $y^2=4 x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O (0,0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$, then which of the following statement($s$) is (are) $TRUE$?

$(A)$ For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $1$

$(B)$ For the ellipse, the eccentricity is $\frac{1}{2}$ and the length of the latus rectum is $\frac{1}{2}$

$(C)$ The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x=1$ is $\frac{1}{4 \sqrt{2}}(\pi-2)$

$(D)$ The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x=1$ is $\frac{1}{16}(\pi-2)$

Tangent is drawn to ellipse $\frac{{{x^2}}}{{27}} + {y^2} = 1$ at $(3\sqrt 3 \cos \theta ,\;\sin \theta )$ where $\theta \in (0,\;\pi /2)$. Then the value of $\theta $ such that sum of intercepts on axes made by this tangent is minimum, is

For the ellipse $\frac{{{x^2}}}{{64}} + \frac{{{y^2}}}{{28}} = 1$, the eccentricity is

The equation of the ellipse whose vertices are $( \pm 5,\;0)$ and foci are $( \pm 4,\;0)$ is