If the straight line $(a-1)x - by + 4 = 0$ is normal to the hyperbola $xy = 1$,then which of the following does not hold?

If $e_1$ and $e_2$ are respectively the eccentricities of the curves $9x^2 - 16y^2 - 144 = 0$ and $9x^2 - 16y^2 + 144 = 0$,then find the value of $\frac{e_1^2 e_2^2}{e_1^2 + e_2^2}$.

If the tangents drawn to the hyperbola $4y^2 = x^2 + 1$ intersect the coordinate axes at the distinct points $A$ and $B$,then the locus of the midpoint of $AB$ is

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

Let the transverse axis of a hyperbola $H$ be parallel to the $X$-axis and $x^2+y^2-2x-4y+3=0$ be the equation of the auxiliary circle of $H$. If the asymptotes of $H$ are at right angles,then the equation of the hyperbola is

The distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2}$. Its equation is