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Coordinate Geometry: Minimum Distance Sum

XAT 2011 · Quantitative Ability & Data Interpretation · Medium · Coordinate Geometry

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In a plane rectangular coordinate system, points L, M, N and O are represented by the coordinates (-5, 0), (1,-1), (0, 5), and (-1, 5) respectively. Consider a variable point P in the same plane. The minimum value of PL + PM + PN + PO is

  1. A.

    1 + \sqrt{37}

  2. B.

    5\sqrt{2} + 2\sqrt{10}

  3. C.

    \sqrt{41} + \sqrt{37}

  4. D.

    \sqrt{41} + 1

  5. E.

    None of these

Answer

B

Explanation

The minimum distance sum PL+PN+PM+POPL + PN + PM + PO for a convex quadrilateral LMNOLMNO is equal to the sum of lengths of the diagonals LN+OMLN + OM. LN=(0−(−5))2+(5−0)2=25+25=52LN = \sqrt{(0 - (-5))^2 + (5 - 0)^2} = \sqrt{25 + 25} = 5\sqrt{2}. OM=(1−(−1))2+(−1−5)2=4+36=40=210OM = \sqrt{(1 - (-1))^2 + (-1 - 5)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10}. Minimum value =52+210= 5\sqrt{2} + 2\sqrt{10}.

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