If the line $2x + 5y = 12$ intersects the ellipse $4x^2 + 5y^2 = 20$ in two distinct points $A$ and $B$,then the mid-point of $AB$ is

The eccentricity of an ellipse whose centre is at the origin is $\frac{1}{2}$. If one of its directrices is $x = -4$,then the equation of the normal to it at $\left(1, \frac{3}{2}\right)$ is

If a tangent having slope of $- \frac{4}{3}$ to the ellipse $\frac{x^2}{18} + \frac{y^2}{32} = 1$ intersects the major and minor axes in points $A$ and $B$ respectively,then the area of $\Delta OAB$ is equal to .................. $sq. \text{ units}$ ($O$ is the centre of the ellipse).

The eccentric angle of the point $(2, \sqrt{3})$ lying on $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ is

Find the position of the point $(4, -3)$ with respect to the ellipse $2x^2 + 5y^2 = 20$.

The equation of the normal to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at the point $(a \cos \theta, b \sin \theta)$ is