The eccentricity of the conic $\frac{5}{r}=2+3 \cos \theta+4 \sin \theta$ is

$A$ circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of the parabola $y^2 = 4ax$. If $PQ$ is the common chord of the circle and the parabola and $L_1L_2$ is the latus rectum,then the area of the trapezium $PL_1L_2Q$ is:

Through the vertex $O$ of the parabola $y^2 = 4ax$,two chords $OP$ and $OQ$ are drawn,and the circles on $OP$ and $OQ$ as diameters intersect in $R$. If $\theta_1, \theta_2$,and $\phi$ are the angles made with the axis by the tangents at $P$ and $Q$ on the parabola and by $OR$ respectively,then the value of $\cot \theta_1 + \cot \theta_2$ is:

If a normal drawn to the ellipse $\frac{x^2}{4} + \frac{y^2}{3} = 1$ touches the hyperbola $\frac{x^2}{4} - \frac{y^2}{3} = 1$,then the square of the slope of that normal is

The foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the hyperbola $\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}$ coincide. Then,the value of $b^2$ is

The angle between the curves $x^2-y^2=4$ and $x^2+y^2=4\sqrt{2}$ is