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Question 30
ln 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
Solution
$$(PL + PN)$$ will be minimum if P lies on LN, and $$(PM + PO)$$ will be minimum if P lies on OM.
=> P must be the intersection point of the diagonals of the quadrilateral.
$$\therefore$$ Min (PL + PM + PN + PO)
= $$LN + OM$$
= $$(\sqrt{(0 + 5)^2 + (5 - 0)^2}) + (\sqrt{(1 + 1)^2 + (-1 - 5)^2})$$
= $$(\sqrt{25 + 25}) + (\sqrt{4 + 36})$$
= $$\sqrt{50} + \sqrt{40} = 5 \sqrt{2} + 2 \sqrt{10}$$
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