If $4x - 3y + k = 0$ touches the ellipse $5x^{2} + 9y^{2} = 45$,then $k$ is equal to

The radius of a circle centered at $(0, 3)$ and passing through the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ is:

The eccentricity of an ellipse,with its centre as origin,is $1/2$. If one of the directrices is $x=4$,then the equation of the ellipse is given by

$A$ tangent is drawn at $(3 \sqrt{3} \cos \theta, \sin \theta)$ $\left(0 < \theta < \frac{\pi}{2}\right)$ to the ellipse $\frac{x^2}{27}+\frac{y^2}{1}=1$. The value of $\theta$ for which the sum of the intercepts on the coordinate axes made by this tangent attains the minimum is

The eccentricity of the curve represented by $x = 3(\cos t + \sin t)$ and $y = 4(\cos t - \sin t)$ is

Let $S$ and $S^{\prime}$ be the foci of an ellipse $E$ and $B$ be one end of its minor axis. Let $\angle S^{\prime} SB = \frac{\pi}{6}$ and $(2 \sqrt{3}, 1)$ be a point on $E$. If $X$-axis is the major axis and $Y$-axis is the minor axis of the ellipse $E$,then the sum of the squares of the lengths of major axis and minor axis is