Hyperbola

If the eccentricity of the standard hyperbola passing through the point $(4,6)$ is $2,$then the equation of the tangent to the hyperbola at $(4,6)$ is:

If a hyperbola passes through the point $P\left(10,16\right)$, and it has vertices at $\left(\pm\,6,0\right)$, then the equation of the normal to it at $P$, is.

If the tangent drawn to the hyperbola $4y^{2}=x^{2}+1$ intersect the co-ordinates axes at the distinct points $A$ and $B$, then the locus of the midpoint of $AB$ is :

Let the hyperbola $H:\frac{x^{2}}{a^{2}}-y^{2}=1$and the ellipse $E:3x^{2}+4y^{2}=12$be such that the length of latus rectum of $H$ is equal to the length of latus rectum of $E$. If $e_{H}$and $e_{E}$are the…

Let$m_{1}$and$m_{2}$be the slopes of the tangents drawn from the point$P\left(4,1\right)$ to the hyperbola$H:\frac{y^{2}}{25}-\frac{x^{2}}{16}=1$If$Q$is the point from which thetangents drawn to$H$have slopes…

If any point on a hyperbola is $(3 \tan \theta, 2 \sec \theta)$, then what is the eccentricity of the hyperbola?

The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through the point $(3 \sqrt{5}, 1)$ and the length of its latus rectum is $\frac{4}{3}$ units. The length of the conjugate axis is

What is the equation of the hyperbola having latus rectum and eccentrieity 8 and $\frac{3}{\sqrt{5}}$ respectively?

A hyperbola has its centre at the origin, passes through the point$(4,2)$and has transverse axis of length$4$along the$x-axis.$Then the eccentricity of the hyperbola is:

A tangent to the hyperbola $\frac{x^2}{4}-\frac{y^2}{2}=1$ meets $x$-axis at $\mathrm{P}$ and $y$-axis at $\mathrm{Q}$. Lines $\mathrm{PR}$ and $\mathrm{QR}$ are drawn such that OPRQ is a rectangle (where $\mathrm{O}$…

Let $H: \dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ be a hyperbola such that the distance between its foci is $6$ and the distance between its directrices is $\dfrac{8}{3}$. If the line $x=\alpha$ intersects the hyperbola $H$…

If the line $\alpha x+2 y=1$, where $\alpha \in \mathbb{R}$, does not meet the hyperbola $x^{2}-9 y^{2}=9$, then a possible value of $\alpha$ is: