The locus of the point of intersection of the lines $(\sqrt{3})kx + ky - 4\sqrt{3} = 0$ and $\sqrt{3}x - y - 4\sqrt{3}k = 0$ is a conic,whose eccentricity is .............

The equation of the hyperbola referred to its axes as axes of coordinate and whose distance between the foci is $16$ and eccentricity is $\sqrt{2}$ is:

At which of the following points is the tangent to the hyperbola $x^2 - y^2 = 3$ parallel to the line $2x + y + 8 = 0$?

If $l$ is the maximum value of $-3x^2+4x+1$ and $m$ is the minimum value of $3x^2+4x+1$,then the equation of the hyperbola having foci at $(l, 0)$ and $(7m, 0)$ and eccentricity $e=2$ is

If the area of the region bounded by $16x^2 - 9y^2 = 144$ and $8x - 3y = 24$ is $A$, then $3(A + 6 \ln(3))$ is equal to . . . . . . .

Let $e_1$ be the eccentricity of a hyperbola for which the distance between its foci is $2$ times the distance between its directrices,and $e_2$ be the eccentricity of another hyperbola for which the length of its transverse axis is twice the length of its conjugate axis. Then $e_1 e_2 =$