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Coordinate Geometry: Line Intersections and Slope

XAT 2019 · Quantitative Ability & Data Interpretation · Medium · Coordinate Geometry

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Let P be the point of intersection of the lines 3x+4y=2a3x + 4y = 2a and 7x+2y=20187x + 2y = 2018 and Q the point of intersection of the lines 3x+4y=20183x + 4y = 2018 and 5x+3y=15x + 3y = 1

If the line through P and Q has slope 2, the value of a is:

  1. A.

    4035

  2. B.

    1/2

  3. C.

    3026

  4. D.

    1

  5. E.

    1009

Answer

C

Explanation

Solving for Q from 3x+4y=20183x + 4y = 2018 and 5x+3y=15x + 3y = 1 gives Q(−550,917)Q(-550, 917). P satisfies 3x+4y=2a3x + 4y = 2a and 7x+2y=20187x + 2y = 2018. Subtracting gives 4x−2y=2018−2a⇒y−2x=a−10094x - 2y = 2018 - 2a \Rightarrow y - 2x = a - 1009. Since line PQ has slope 2, y−2xy - 2x is constant for all points on PQ. For Q(-550, 917): yQ−2xQ=917−2(−550)=2017y_Q - 2x_Q = 917 - 2(-550) = 2017. Thus a−1009=2017⇒a=3026a - 1009 = 2017 \Rightarrow a = 3026.

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