JEE Previous Year Question

For $a>0$, let the curves $C_1\text{:}{y}^2=\text{ax}$ and $C_2\text{:}{x}^2=\text{ay}$ intersect at origin $O$ and a point $P$. Let the line $x=b\text{(}0<b<a\text{)}$ intersect the chord $OP$ and the $x$ -axis at points $Q$ and $R$, respectively. If the line $x=b$ bisects the area bounded by the curves, $C_1$ and $C_2$, and the area of $\Delta \text{OQR}=\frac{1}{2}$, then $a$ satisfies the equation:

Topic: JEE PYQ

Answer: C. Full step-by-step solution below.