The value of $k$ such that the line $y=2x+k$ touches the ellipse $3x^2+5y^2=15$ is

An ellipse has $OB$ as semi-minor axis,$S$ and $S^{\prime}$ are foci and angle $\angle SBS^{\prime}$ is a right angle. Then the eccentricity of the ellipse is

Find the equation of the ellipse with its center at the origin,passing through the points $(-3, 1)$ and $(2, -2)$,given that $a > b$.

In an ellipse,the minor axis is $8$ and the eccentricity is $\frac{\sqrt{5}}{3}$. Then the major axis is:

If $(l, m)$ is the circumcentre of an equilateral triangle inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ having vertices at points with eccentric angles $\theta_1, \theta_2$ and $\theta_3$,then $\frac{2}{3}\left[\cos \left(\theta_1-\theta_2\right)+\cos \left(\theta_2-\theta_3\right)+\cos \left(\theta_3-\theta_1\right)\right]=$

The centre of an ellipse is $C$,$PN$ is any ordinate,and $A$,$A'$ are the end points of the major axis. Then the value of $\frac{PN^2}{AN \cdot A'N}$ is