Explanation

(d)
The equation of hyperbola is
$4x^{2} - 9y^{2} = 36$
$\Rightarrow \frac{x^{2}}{9} - \frac{y^{2}}{4} = 1$ …(i)
The equation of the chords of contact of tangents from $(x_{1},y_{1})$ and $(x_{2},\ y_{2})$ to the given hyperbola are
$\frac{x{\ x}_{1}}{9} - \frac{y{\ y}_{1}}{4} = 1$ …(ii)
and $\ \frac{x{\ x}_{2}}{9} - \frac{y{\ y}_{2}}{4} = 1$ …(iii)
Lines (ii) and (iii) are at right angles.
$\therefore\frac{9}{4}.\frac{x_{1}}{y_{1}} \times \frac{4}{9}.\frac{x_{2}}{y_{2}} = - 1$
$\Rightarrow \ \frac{x_{1}x_{2}}{y_{1}y_{2}} = - \left( \frac{9}{4} \right)^{2} = - \frac{81}{16}$