$A$ hyperbola with centre at $(0,0)$ has its transverse axis along the $X$-axis,and its length is $12$. If $(8,2)$ is a point on the hyperbola,then its eccentricity is

$A$ hyperbola having the transverse axis of length $\sqrt{2}$ has the same foci as that of the ellipse $3x^{2} + 4y^{2} = 12$. Then this hyperbola does not pass through which of the following points?

The eccentricity of the hyperbola $4x^2 - 9y^2 = 16$ is

$P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$ are two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where $\phi+\theta=\frac{\pi}{2}$. If $(h, k)$ is the point of intersection of the normals drawn at $P$ and $Q$,then $k=$

If the product of the lengths of the perpendiculars from any point on the hyperbola $16x^2 - 25y^2 = 400$ to its asymptotes is $p$ and the angle between the two asymptotes is $\theta$,then $p \tan \frac{\theta}{2} =$

The line $2x + y = 1$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > b)$. If this line passes through the point of intersection of a directrix and the positive $X$-axis,then the eccentricity of that hyperbola is