The eccentricity of the conic section $x^2 - 4y^2 = 1$ is:

If $e_1$ is the eccentricity of the ellipse $\frac{x^2}{16}+\frac{y^2}{25}=1$ and $e_2$ is the eccentricity of a hyperbola passing through the foci of the given ellipse and $e_1 e_2=1$,then the equation of such a hyperbola among the following is

The difference of the focal distances of any point on the hyperbola $9x^2 - 16y^2 = 144$ is

The equation of the normal to the hyperbola $\frac{x^2}{25} - \frac{y^2}{16} = 1$ that is perpendicular to the line $2x + y = 1$ is:

$PQ$ is a double ordinate of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ such that $\triangle OPQ$ is an equilateral triangle,where $O$ is the centre of the hyperbola. Then the eccentricity $e$ of the hyperbola satisfies:

Let $H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,where $a > b > 0$,be a hyperbola in the $xy$-plane whose conjugate axis $LM$ subtends an angle of $60^{\circ}$ at one of its vertices $N$. Let the area of the triangle $LMN$ be $4\sqrt{3}$.

List-$I$	List-$II$
$P$. The length of the conjugate axis of $H$ is	$1$. $8$
$Q$. The eccentricity of $H$ is	$2$. $\frac{4}{\sqrt{3}}$
$R$. The distance between the foci of $H$ is	$3$. $\frac{2}{\sqrt{3}}$
$S$. The length of the latus rectum of $H$ is	$4$. $4$

The correct option is: