The angle between the curves $2x^2 + y^2 = 20$ and $4y^2 - x^2 = 8$ at a point where they intersect in the $4^{th}$ quadrant is

The equation of the line passing through the points of intersection of the parabola $x^2 = 8y$ and the ellipse $\frac{x^2}{3} + y^2 = 1$ is

The square of the slope of a common tangent drawn to the circle $4x^2 + 4y^2 = 25$ and the ellipse $4x^2 + 9y^2 = 36$ is

Let $P(x_1, y_1)$ and $Q(x_2, y_2)$,with $y_1 < 0$ and $y_2 < 0$,be the endpoints of the latus rectum of the ellipse $x^2 + 4y^2 = 4$. The equations of the parabolas with latus rectum $PQ$ are:
$(A) x^2 + 2\sqrt{3}y = 3 + \sqrt{3}$
$(B) x^2 - 2\sqrt{3}y = 3 + \sqrt{3}$
$(C) x^2 + 2\sqrt{3}y = 3 - \sqrt{3}$
$(D) x^2 - 2\sqrt{3}y = 3 - \sqrt{3}$

The locus of the midpoints of the chords of the hyperbola $x^2 - y^2 = a^2$ which are tangents to the parabola $x^2 = 4by$ will be -

The quadratic equation whose roots are $l$ and $m$,where $l = \lim_{\theta \rightarrow 0} \left( \frac{3 \sin \theta - 4 \sin^2 \theta}{\theta} \right)$ and $m = \lim_{\theta \rightarrow 0} \frac{2 \tan \theta}{\theta(1 - \tan^2 \theta)}$,is: