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

Slope of common tangents of parabola $(x -1)^2 = 4(y -2)$ and ellipse ${\left( {x - 1} \right)^2} + \frac{{{{\left( {y - 2} \right)}^2}}}{2} = 1$ are $m_1$ and $m_2$ ,then $m_1^2 + m_2^2$ is equal to

The equation $\frac{{{x^2}}}{{2 - r}} + \frac{{{y^2}}}{{r - 5}} + 1 = 0$ represents an ellipse, if

An ellipse and a hyperbola have the same centre origin, the same foci and the minor-axis of the one is the same as the conjugate axis of the other. If $ e_1, e_2 $ be their eccentricities respectively, then  $e_1^{ - 2} + e_2^{ - 2}$ equals

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

If the distance between a focus and corresponding directrix of an ellipse be $8$ and the eccentricity be $1/2$, then length of the minor axis is