The equation of the tangent at the point $(1/4, 1/4)$ of the ellipse $\frac{x^2}{4} + \frac{y^2}{12} = 1$ is

If $A$ and $B$ are two fixed points and $P$ is a variable point such that $PA + PB = 4$,then the locus of $P$ is a/an

If the distance between the foci of an ellipse is equal to the length of the latus rectum, then its eccentricity is

An ellipse having the coordinate axes as its axes and its major axis along the $Y$-axis,passes through the point $(-3, 1)$ and has eccentricity $e = \sqrt{\frac{2}{5}}$. Then its equation is:

If the distance between the directrices is thrice the distance between the foci,then the eccentricity of the ellipse is

Consider the curve $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of