The locus of the middle points of the chords of hyperbola $3{x^2} - 2{y^2} + 4x - 6y = 0$ parallel to $y = 2x$ is

The condition that the straight line $lx + my = n$ may be a normal to the hyperbola ${b^2}{x^2} - {a^2}{y^2} = {a^2}{b^2}$ is given by

The equation to the chord joining two points $(x_1, y_1)$  and $(x_2, y_2)$  on the rectangular hyperbola $xy = c^2$  is

The point of contact of the line $y = x - 1$ with $3{x^2} - 4{y^2} = 12$ is

Locus of foot of normal drawn from any focus to variable tangent of hyperbola $4x^2-9y^2-8x- 18y = 41$ will be

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 -