An Equidistant Tangency

Consider the ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{3y^2}{a^2} = 1$ (with $a > 0$), and points $P, Q, R$ on it such that the distances from $R$ to the lines $OP$ and $OQ$ both equal $\dfrac{1}{2}a$. Let $k_1, k_2$ be the slopes of $OP, OQ$.

1. Find $k_1k_2$. 2. Prove that $|OP|\cdot|OQ| \leqslant \dfrac{2}{3}a^2$.