The length of the latus rectum of an ellipse is $6$ units and the distance between a focus and its nearest vertex on the major axis is $\frac{5}{3}$ units. If $e$ is the eccentricity of this ellipse,then $e$ satisfies the equation:

$A$ tangent having slope of $-\frac{4}{3}$ to the ellipse $\frac{x^2}{18} + \frac{y^2}{32} = 1$ intersects the major and minor axes in points $A$ and $B$ respectively. If $C$ is the centre of the ellipse,then the area of the triangle $ABC$ is: .............. $sq. \,units$

The locus of the midpoints of the intercepted portion of the tangents by the coordinate axes,which are drawn to the ellipse $x^2+2y^2=2$,is

Let the common tangents to the curves $4(x^{2}+y^{2}) = 9$ and $y^{2} = 4x$ intersect at the point $Q$. Let an ellipse,centered at the origin $O$,have lengths of semi-minor and semi-major axes equal to $OQ$ and $6$,respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse,then $\frac{l}{e^{2}}$ is equal to

The line $y = 2t^2$ intersects the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ in real points if

If the eccentricities of the two ellipses $\frac{x^2}{169} + \frac{y^2}{25} = 1$ and $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ are equal,then the value of $a/b$ is