The equation of the normal to the curve $3x^2 - y^2 = 8$,which is parallel to the line $x + 3y = 10$,is

Let $PQ$ be a chord of the hyperbola $\frac{x^2}{4} - \frac{y^2}{b^2} = 1$, perpendicular to the $x$-axis such that $OPQ$ is an equilateral triangle, where $O$ is the centre of the hyperbola. If the eccentricity of the hyperbola is $\sqrt{3}$, then the area of the triangle $OPQ$ is:

Let $H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be a hyperbola such that the distance between its foci is $6$ and the distance between its directrices is $\frac{8}{3}$. If the line $x = k$ intersects the hyperbola $H$ at the points $A$ and $B$ such that the area of the triangle $AOB$ is $4\sqrt{15}$, where $O$ is the origin, then $a^2$ equals

The tangents drawn to the hyperbola $5x^2 - 9y^2 = 90$ through a variable point $P$ make the angles $\alpha$ and $\beta$ with its transverse axis. If $\alpha$ and $\beta$ are complementary angles,then the locus of $P$ is

The normal to the rectangular hyperbola $xy = c^2$ at the point $t_1$ meets the curve again at the point $t_2$. Then the value of $t_1^3 t_2$ is

The length of the latus rectum of the curve $xy = 7x + 5y$ is