The locus of the point of intersection of the lines $bxt - ayt = ab$ and $bx + ay = abt$ is

The equation to the hyperbola having its eccentricity $2$ and the distance between its foci is $8$

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 .............

If the two tangents drawn on hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ in such a way that the product of their gradients is ${c^2}$, then they intersects on the curve

The locus of the point of intersection of any two perpendicular tangents to the hyperbola is a circle which is called the director circle of the hyperbola, then the eqn of this circle is

The point of contact of the line $y = x - 1$ with $3{x^2} - 4{y^2} = 12$ is