The number of values of $c$ such that the line $y = cx + c$,where $c \in R$,touches the ellipse $\frac{x^2}{4} + \frac{y^2}{1} = 1$ is:

Let $A_1, A_2, A_3$ be regions in the $XY$-plane defined by:
$A_1 = \{(x, y) : x^2 + 2y^2 \leq 1\}$
$A_2 = \{(x, y) : |x|^3 + 2\sqrt{2}|y|^3 \leq 1\}$
$A_3 = \{(x, y) : \max(|x|, \sqrt{2}|y|) \leq 1\}$
Then,

Let $S=\left\{(x, y) \in N \times N : 9(x-3)^{2}+16(y-4)^{2} \leq 144\right\}$ and $T=\left\{(x, y) \in R \times R :(x-7)^{2}+(y-4)^{2} \leq 36\right\}$. Then $n(S \cap T)$ is equal to $......$

The line $12 x \cos \theta + 5 y \sin \theta = 60$ is tangent to which of the following curves?

The equation of the ellipse whose centre is at the origin and which passes through the points $(-3, 1)$ and $(2, -2)$ is

$A$ focus of an ellipse is at the origin. The directrix is the line $x = 4$ and the eccentricity is $\frac{1}{2}$. Then the length of the semi-major axis is