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geometry problem

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Coordinate Geometry

Harbin, 2019

In the Cartesian plane, O is the origin. The line y = 43x + 4 meets the x-axis at A and the y-axis at B. Line BC meets the x-axis at C, where C is the reflection of A in the y-axis.

  1. Find the equation of line BC.
  2. Let P lie on segment AB and Q on segment BC, with BQ = AP. Join PQ. Let the x-coordinate of P be t, and let the area of △PBQ be S, where S ≠ 0. Express S as a function of t. (You do not need to state the domain.)
  3. Under the conditions in (2), let E lie on segment OA, and let R lie on BC extended beyond C, with y-coordinate −25. Draw PE, BE, and AQ. Let F = AQ ∩ BE, with ∠APE = ∠CBE. Draw PF and extend it to meet the negative y-axis at M. Join QM and MR. If tan ∠QMR = 2423, find the equation of line PM.
Original coordinate diagram: A on the negative x-axis, B on the positive y-axis, C on the positive x-axis, and origin O. Lines AB and BC form a triangle. Original construction diagram with P on AB, Q on BC, E on OA, F at AQ intersect BE, M on the negative y-axis, and R beyond C. Segments PE, BE, AQ, PM, QM, and MR are drawn.

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