If the coordinates of a point are given by the equations $x = b \sec \phi$ and $y = a \tan \phi$,then its locus is

If the equation of the tangent to the hyperbola $5x^2 - 9y^2 - 20x - 18y - 34 = 0$ which makes an angle of $45^{\circ}$ with the positive $X$-axis is $x + by + c = 0$,then $b^2 + c^2 =$

The midpoint of the chord $4x - 3y = 5$ of the hyperbola $2x^2 - 3y^2 = 12$ is

The locus of the mid-point of the chord of the hyperbola $3x^2 - 2y^2 + 4x - 6y = 0$ which is parallel to $y = 2x$ is:

$A$ mirror in the first quadrant is in the shape of a hyperbola whose equation is $xy=1$. $A$ light source in the second quadrant emits a beam of light that hits the mirror at the point $(2, 1/2)$. If the reflected ray is parallel to the $Y$-axis,the slope of the incident beam is

If the angle between the asymptotes of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $2 \tan^{-1}\left(\frac{b}{a}\right) = 2 \tan^{-1}\left(\frac{2}{3}\right)$ and $a^2-b^2=45$,then $ab=$