If the curves $y^2=6x$ and $9x^2+by^2=16$ intersect each other at right angles,then the value of $b$ is

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:

The locus of the foot of the perpendicular drawn from the centre upon any tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is

Consider the circle $x^2+y^2=9$ and the parabola $y^2=8x$. They intersect at $P$ and $Q$ in the first and the fourth quadrants,respectively. Tangents to the circle at $P$ and $Q$ intersect the $x$-axis at $R$ and tangents to the parabola at $P$ and $Q$ intersect the $x$-axis at $S$.
$1.$ The ratio of the areas of the triangles $PQS$ and $PQR$ is
$(A)$ $1:\sqrt{2}$ $(B)$ $1:2$ $(C)$ $1:4$ $(D)$ $1:8$
$2.$ The radius of the circumcircle of the triangle $PRS$ is
$(A)$ $5$ $(B)$ $3\sqrt{3}$ $(C)$ $3\sqrt{2}$ $(D)$ $2\sqrt{3}$
$3.$ The radius of the incircle of the triangle $PQR$ is
$(A)$ $4$ $(B)$ $3$ $(C)$ $8/3$ $(D)$ $2$
Give the answer for questions $1, 2$ and $3.$

Let the eccentricity of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be the reciprocal of the eccentricity of the hyperbola $2x^2 - 2y^2 = 1$. If the ellipse intersects the hyperbola at right angles,then the square of the length of the latus-rectum of the ellipse is $................$.

Find the area of the region enclosed by the equation $2|x| + 3|y| = 6$ in square units.