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

If the length and breadth of a rectangle of maximum area that can be inscribed in an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are $8 \sqrt{2}$ and $4 \sqrt{2}$ respectively,then the eccentricity of that ellipse is

The length of the chord of the ellipse $\frac{x^2}{4} + y^2 = 1$ formed on the line $y = x + 1$ is

$P$ is a point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. When the area of $\Delta PSS'$ is maximum,the inradius of $\Delta PSS'$ ($S$ and $S'$ are foci) is equal to:

If $a$ and $c$ are positive real numbers and the ellipse $\frac{x^2}{4c^2} + \frac{y^2}{c^2} = 1$ has four distinct points in common with the circle $x^2 + y^2 = 9a^2$,then

Tangents are drawn from point $(1, 1)$ to the ellipse $S \equiv x^2 + 4y^2 - 2x + 8y + 1 = 0$. If $m_1, m_2$ $(m_1 > m_2)$ are the slopes of these tangents,then with respect to the given ellipse,the point $P(m_1, m_2)$: