Chord of contact subtends right angle at origin implies angle between tangents is $90^\circ$
Let $T = 0$ be the chord of contact. For conic, if tangents from a point $(x_1, y_1)$ are perpendicular, then:
$S_1 = \dfrac{1}{2} S$ where $S$ is the equation of conic and $S_1$ its value at point
For hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$, condition becomes:
$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = \dfrac{1}{2}$