The curve $xy = c, (c > 0)$, and the circle $x^2 + y^2 = 1$ touch at two points. Then the distance between the points of contacts is

Product of length of the perpendiculars drawn from foci on any tangent to hyperbola ${x^2} – \frac{{{y^2}}}{4}$ = $1$ is

If line $ax$ + $by$ = $1$ is normal to the hyperbola $\frac{{{x^2}}}{{{p^2}}} – \frac{{{y^2}}}{{{q^2}}} = 1$ then $\frac{{{p^2}}}{{{a^2}}} – \frac{{{q^2}}}{{{b^2}}} = 1$ is equal to (where $a$,$b$,$p$, $q \in {R^ + })$-

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

The foci of the ellipse $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{{b^2}}} = 1$ and the hyperbola $\frac{{{x^2}}}{{144}} – \frac{{{y^2}}}{{81}} = \frac{1}{{25}}$ coincide. Then the value of $b^2$ is –