The locus of a point which moves such that the difference of its distances from two fixed points is always a constant is

If $P(x_1, y_1)$ is a point on the hyperbola $x^2 - y^2 = a^2$,then $SP \cdot S'P = \_\_\_\_$

The equation of the hyperbola whose directrix is $x + 2y = 1$,focus $(2, 1)$ and eccentricity $e = 2$ is:

The locus of the point of intersection of the lines $\sqrt{3}x - y - 4\sqrt{3}t = 0$ and $\sqrt{3}tx + ty - 4\sqrt{3} = 0$ (where $t$ is a parameter) is a hyperbola whose eccentricity is

If $P(\theta)$ lies on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ and $S$ and $S^{\prime}$ are the foci of the hyperbola,then $SP \cdot S^{\prime}P =$

The equation of one of the tangents drawn from the point $(0, 1)$ to the hyperbola $45x^2 - 4y^2 = 5$ is