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Conic Sections
Parabola
Hard

Question

Let PQ\mathrm{PQ} be a focal chord of the parabola y2=36xy^{2}=36 x of length 100 , making an acute angle with the positive xx-axis. Let the ordinate of P\mathrm{P} be positive and M\mathrm{M} be the point on the line segment PQ such that PM : MQ = 3 : 1. Then which of the following points does NOT lie on the line passing through M and perpendicular to the line PQ\mathrm{PQ}?

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Solution

This problem involves several key concepts from coordinate geometry, specifically related to parabolas, focal chords, section formula, and properties of perpendicular lines. We will systematically break down the problem to find the required line and then check the given points.

1. Analyze the Parabola's Equation and Determine Key Parameters

  • Key Concept: The standard form of a parabola y2=4axy^2 = 4ax helps us identify its focal length and the coordinates of its focus.
  • Step 1.1: Determine the focal length 'a' and the focus. The given equation of the parabola is y2=36xy^2 = 36x. Explanation: We compare this to the standard form y2=4axy^2 = 4ax.
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