The locus of the point of intersection of lines $(x + y)t = a$ and $x - y = at$,where $t$ is the parameter,is

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=b^2$ at four points $(x_1, y_1)$,$(x_2, y_2)$,$(x_3, y_3)$,and $(x_4, y_4)$,then $y_1 y_2 y_3 y_4 = $

Let the hyperbola be $x^2 - 2y^2 - 2\sqrt{2}x - 4\sqrt{2}y - 6 = 0$. Let $A$ be a vertex of the hyperbola. Let $B$ be an endpoint of the latus rectum near $A$. If $C$ is the focus of the hyperbola nearest to $A$,find the area of triangle $ABC$.

If $l_1$ and $l_2$ are the lengths of the perpendiculars drawn from a point on the hyperbola $5x^2 - 4y^2 - 20 = 0$ to its asymptotes,then $\frac{l_1^2 l_2^2}{100} = $

The equation $\frac{1}{r} = \frac{1}{8} + \frac{3}{8} \cos \theta$ represents:

$A$ mirror in the first quadrant is in the shape of a hyperbola whose equation is $xy=1$. $A$ light source in the second quadrant emits a beam of light that hits the mirror at the point $(2, 1/2)$. If the reflected ray is parallel to the $Y$-axis,the slope of the incident beam is