Let $O$ be the vertex of a parabola and $Q$ be any point on the axis of the parabola. If $PQR$ be any chord passing through $Q$ and $PM$ and $RN$ be the ordinates of $P$ and $R$, then:
Step-by-Step Solution
Key Concept: When a focal chord passes through a fixed point on the axis, the products of parametric distances satisfy $OM \cdot ON = OQ^2$ and $PM \cdot RN = 4aOQ$.
For parabola $y^2 = 4ax$ with $P(at_1^2, 2at_1)$ and $R(at_2^2, 2at_2)$, the chord $PR$ has equation $(t_1 + t_2)y = 2x + 2at_1t_2$. If this passes through $Q(h, 0)$, then $2h = 2at_1t_2$, so $t_1t_2 = \frac{h}{a}$. Computing $OM \cdot ON = at_1^2 \cdot at_2^2 = a^2(t_1t_2)^2 = a^2\left(\frac{h}{a}\right)^2 = h^2 = OQ^2$ and $PM \cdot RN = |2at_1| \cdot |2at_2| = 4a^2|t_1t_2| = 4ah = 4aOQ$, establishing the required geometric relations.
Correct Answer: 1,3