Eccentricity of the curve ${x^2} - {y^2} = {a^2}$ is

The graph of the conic $ x^2 - (y - 1)^2 = 1$  has one tangent line with positive slope that passes through the origin. the point of tangency being $(a, b). $ Then  Length of the latus rectum of the conic is

The equation to the hyperbola having its eccentricity $2$ and the distance between its foci is $8$

The eccentricity of the hyperbola ${x^2} - {y^2} = 25$ is

Let $P (3\, sec\,\theta , 2\, tan\,\theta )$ and $Q\, (3\, sec\,\phi , 2\, tan\,\phi )$ where $\theta + \phi \, = \frac{\pi}{2}$ , be two distinct points on the hyperbola $\frac{{{x^2}}}{9} - \frac{{{y^2}}}{4} = 1$ . Then the ordinate of the point of intersection of the normals at $P$ and $Q$ is

$P(6, 3)$ is a point on the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ . If the normal at point $P$ intersect the $x-$ axis at $(10, 0)$ , then the eccentricity of the hyperbola is