If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $(x_i, y_i)$,for $i=1, 2, 3, 4$,then $y_1+y_2+y_3+y_4$ equals

If the latus rectum of a hyperbola through one focus subtends an angle of $60^{\circ}$ at the other focus,then its eccentricity is

If the perimeter of a triangle is $20$ and two of its vertices are $(-5, 0)$ and $(6, 0)$,then the locus of the third vertex is:

The tangent drawn at an extremity (in the first quadrant) of the latus rectum of the hyperbola $\frac{x^2}{4}-\frac{y^2}{5}=1$ meets the $x$-axis and $y$-axis at $A$ and $B$ respectively. If $O$ is the origin,then $(OA)^2-(OB)^2=$

The locus of a point of intersection of two lines $x \sqrt{3}-y=k \sqrt{3}$ and $\sqrt{3} k x+k y=\sqrt{3}$,where $k \in R$,describes:

The eccentricity of the hyperbola $\frac{\sqrt{1999}}{3}(x^2 - y^2) = 1$ is