The number of values of $c$ such that the line $y=4x+c$ touches the curve $\frac{x^{2}}{4}+y^{2}=1$ is

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

$A$ focus of an ellipse having eccentricity $e = \frac{1}{2}$ is at $(0,0)$ and a directrix is the line $x = 4$. Then the equation of one such ellipse is

If the normal drawn at the point $P(\frac{\pi}{4})$ on the ellipse $x^2+4y^2-4=0$ meets the ellipse again at $Q(\alpha, \beta)$,then $\alpha=$

Planet $M$ orbits around its sun,$S$,in an elliptical orbit with the sun at one of the foci. When $M$ is closest to $S$,it is $2$ units away. When $M$ is farthest from $S$,it is $18$ units away. Assuming $S$ is at the origin $(0, 0)$ and the other focus lies on the negative $y$-axis,find the equation of the elliptical orbit of planet $M$.

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 ...