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

Let $S$ denote the locus of the point of intersection of the pair of lines $4x - 3y = 12\alpha$ and $4\alpha x + 3\alpha y = 12$, where $\alpha$ varies over the set of non-zero real numbers. Let $T$ be the tangent to $S$ passing through the points $(p, 0)$ and $(0, q)$, with $q > 0$, and parallel to the line $4x - \frac{3}{\sqrt{2}}y = 0$. Then the value of $pq$ is (in $\sqrt{2}$)

Find the equation of the hyperbola satisfying the given conditions: Foci $(\pm 3 \sqrt{5}, 0)$,the latus rectum is of length $8$.

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

The equation of the common tangents to the two hyperbolas $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ is:

Let $P(3,3)$ be a point on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. If the normal to it at $P$ intersects the $x$-axis at $(9,0)$ and $e$ is its eccentricity,then the ordered pair $(a^{2}, e^{2})$ is equal to