JEE-Advanced 2015 (12/40)

Geometry Level 4

In \(\mathbb{R}^3\), let \(L\) be a straight line passing through the origin. Suppose that all the points on \(L\) are at a constant distance from the two planes \(P_1:x+2y-z+1=0\) and \(P_2:2x-y+z-1=0\). Let \(M\) be the locus of the feet of the perpendiculars drawn fro the points on \(L\) to the plane \(P_1\). Which of the following points lie(s) on \(M\) ?
\[\begin{array}{} (1) \, \left(0,-\frac{5}{6},-\frac{2}{3} \right) \quad \quad \quad \quad & (2) \, \left(-\frac{1}{6},-\frac{1}{3},\frac{1}{6} \right) \\ (3) \, \left( -\frac{5}{6},0,\frac{1}{6} \right) & (4) \, \left( -\frac{1}{3},0,\frac{2}{3} \right) \end{array}\]
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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