Tangent to a Hyperbola

If the line $y=mx+\sqrt{a^2{m}^2-b^2}, m=\frac{1}{2}$ touches the hyperbola $\frac{x^2}{16}-\frac{y^2}{3}=1$ at the point $\left(4sec\theta , \sqrt{3}\tan \theta \right)$ then $\theta$ is

Find equations of the tangents drawn from $(-4,-3)$ to hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=-1$.

Find equations of the tangents drawn from $(4,-3)$ to hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=-1$.

The locus of the point of intersection of two tangents of the hyperbola $\frac{x^2}{2}-\frac{y^2}{4}=1,$ if the product of their slopes is $1,$ is

The number of possible tangents to the curve $9x^2 –16y^2=144$, which are also the normal to circle $x^2+ y^2–5=4x+2y$, is

The equation of tangent to the hyperbola, $4x^2-5y^2=20$, which is parallel to $x-y=2$, is

If the line $2x+\sqrt{6}y=2$ touches the hyperbola $x^2-2y^2=4,$ then the point of contact is

The equation of tangents to the hyperbola $3x^2-2y^2=6$ , which is perpendicular to the line $x-3y=3$ , are

The values of the $m$ for which the line $y=mx+2$ becomes a tangent to the hyperbola $4x^2-9y^2=36$ is

The product of length of perpendicular from any point on the hyperbola $x^2-y^2=16$ to is symptoes is

Asymptotes of Hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$ are .....

Asymptotes of Hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$ are .....

From any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, tangents are drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=2$. The area cut off by the chord of contact on the region between the asymptotes is equal to -

Find the hyperbola whose asymptotes are $2x-y=3 and 3x+y-7=0$ and which passes through the point $(1, 1)$:

The equation of the tangent to the hyperbola $3x^2-y^2=3$ which is perpendicular to the line $x+3y+3=0$ is

The equation of the tangent to the hyperbola $3x^2-y^2=3$, which is perpendicular to the line $x+3y-2=0$ is

The product of the perpendicular from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ to its asymptotes is equal to

If the ordinate of any point $P$ on the hyperbola $9x^2-16y^2=144$, is produced to cut the asymptotes in the points $Q$ and $R$,then the product $PQ.PR$ equals to

The product of the lengths of perpendiculars drawn from any point on the hyperbola $x^2-2y^2-2=0$, to its asymptotes is

The equation of the tangent parallel to $y-x+5=0$, drawn to $\frac{x^2}{3}-\frac{y^2}{2}=1$ is