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

Let $L$ be a common tangent line to the curves $4 x^{2}+9 y^{2}=36$ and $(2 x)^{2}+(2 y)^{2}=31$. Then the square of the slope of the line $L$ is ..... .

If the line $y = 2x + c$ be a tangent to the ellipse $\frac{{{x^2}}}{8} + \frac{{{y^2}}}{4} = 1$, then $c = $

The line $y = mx + c$ is a normal to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1$, if $c = $

The equation of ellipse whose distance between the foci is equal to $8$ and distance between the directrix is $18$, is

An ellipse inscribed in a semi-circle touches the circular arc at two distinct points and also touches the bounding diameter. Its major axis is parallel to the bounding diameter. When the ellipse has the maximum possible area, its eccentricity is