The distance between the directrices of the conic $\sqrt{(x - 5)^2 + (y - 4)^2} + \sqrt{(x - 3)^2 + (y - 2)^2} = 6$ is

Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{2}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b)$ be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is

In an ellipse,the distance between its foci is $6$ and the length of the major axis is $8$. Find its eccentricity.

If $P$ is any point on the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$ and $S$ and $S^{\prime}$ are the foci,then $PS + PS^{\prime}$ is equal to

If $S$ and $S^{\prime}$ are the foci of an ellipse $\frac{x^2}{169}+\frac{y^2}{144}=1$ and the point $B$ lying on the positive $Y$-axis is one end of its minor axis,then the incenter of the triangle $SBS^{\prime}$ is

If the foci of an ellipse are $(\pm \sqrt{5}, 0)$ and its eccentricity is $\frac{\sqrt{5}}{3}$,then the equation of the ellipse is