Geometry Level 3

Let $$P$$ and $$Q$$ be distinct points on the parabola $$y^2=2x$$ such that a circle with $$PQ$$ as diameter passes through the vertex $$O$$ of the parabola. If $$P$$ lies in the first quadrant and the area of the triangle $$\Delta OPQ$$ is $$3\sqrt{2}$$, then which of the following is(are) the coordinates of $$P$$ ?

$\begin{array}{} (1) \, (4,2\sqrt{2}) \quad \quad \quad \quad & (2) \, (9,3\sqrt{2}) \\ (3) \, (\frac{1}{4}, \frac{1}{\sqrt{2}} ) & (4) \, (1,\sqrt{2}) \end{array}$
Note :

• Submit your answer as the increasing order of the serial numbers of all the correct options.

• For eg, if your answer is $$(1),(2)$$, then submit 12 as the correct answer, if your answer is $$(2),(3),(4)$$, then submit 234 as the correct answer.

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