The equation of the ellipse having a vertex at $(6,1)$,a focus at $(4,1)$ and the eccentricity $e = \frac{3}{5}$ is

The ratio of the distance of a point from a fixed point and a line $x = 9/2$ is always $2 : 3$. Then the locus of the point is

The centre of a circle $C$ is at the centre of the ellipse $E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b$. Let $C$ pass through the foci $F_1$ and $F_2$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let $P$ be one of these four points. If the area of the triangle $PF_1F_2$ is $30$ and the length of the major axis of $E$ is $17$,then the distance between the foci of $E$ is:

Let the eccentricity of the ellipse $2x^2 + ay^2 - 8x - 2ay + (8 - a) = 0$ be $\frac{1}{\sqrt{3}}$. If the major axis of this ellipse is parallel to the $Y$-axis,then the equation of the tangent to this ellipse with slope $1$ is:

The angle between the tangents drawn from the point $(1, 2)$ to the ellipse $3x^2 + 2y^2 = 5$ is

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