Question

Sol. [Key Concept: A chord subtending a right angle at $O$: homogenise the conic with the line ⇒ coeff $x^{2}$ + coeff $y^{2 = 0}$.]

Homogeneous equation of the curve with line.

Coefficient of $x^{2} +$ coefficient of $y^{2} = 0$

homogenize the curve with the chord $ax + by = 1$:

$${px}^{2 + 2}axy + {qy}^{2 =}r(ax + by)^{2}$$

$$\Rightarrow \ (p - {ra}^{2)}x^{2 + 2(}a - rab)xy + (q - {rb}^{2)}y^{2 = 0}$$

This pair $OA,OB$ subtends a right angle at $O$

⇒ coeff $x^{2}$ + coeff $y^{2 = 0}$:

$(p - {ra}^{2) + (}q - {rb}^{2) = 0} \Rightarrow \ r(a^{2 +}b^{2) =}p + q$ → (a).