The-plan-is-provided-with-an-orthonormal-reference-O-I-J-the-following-points-are-given-A-1-2-B-2-3-C-1-9-We-assume-that-the-point-O-is-the-barycenter-of-the-point-A-B-C-O-bar-A-3-B-1-

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Question Number 77356 by mathocean1 last updated on 05/Jan/20

$$\mathrm{The}\mathrm{plan}\mathrm{is}\mathrm{provided}\mathrm{with}\mathrm{an}$$$$\mathrm{orthonormal}\mathrm{reference}(\mathrm{O}.\mathrm{I}.\mathrm{J}).$$$$\mathrm{the}\mathrm{following}\mathrm{points}\mathrm{are}\mathrm{given}$$$$\mathrm{A}(1,2)\mathrm{B}(-2,3)\mathrm{C}(1,9).$$$$\mathrm{We}\mathrm{assume}\mathrm{that}\mathrm{the}\mathrm{point}\mathrm{O}\mathrm{is}\mathrm{the}$$$$\mathrm{barycenter}\mathrm{of}\mathrm{the}\mathrm{point}\mathrm{A},\mathrm{B},\mathrm{C}.$$$$\to \mathrm{O}=\mathrm{bar}\{(\mathrm{A};3),(\mathrm{B};1),(\mathrm{C};-1)\}$$$$$$$$\mathrm{Question}1$$$$\mathrm{knowing}\mathrm{that}$$$${3\mathrm{MA}}^2+{\mathrm{MB}}^2-{\mathrm{MC}}^2={3\mathrm{MO}}^2+{3\mathrm{OA}}^2+{\mathrm{OB}}^2-{\mathrm{OC}}^2$$$$\mathrm{Determine}\mathrm{and}\mathrm{construct}\mathrm{the}\mathrm{set}$$$$\mathrm{of}\mathrm{points}\mathrm{M}\mathrm{on}\mathrm{the}\mathrm{plane}\mathrm{such}\mathrm{as}$$$${3\mathrm{MA}}^2+{\mathrm{MB}}^2-{\mathrm{MC}}^2=-42$$$$$$$$$$$$$$

Answered by mind is power last updated on 05/Jan/20

$${3\mathrm{MA}}^2+{\mathrm{MB}}^2-{\mathrm{MC}}^2={3\mathrm{MO}}^2+{3\mathrm{OA}}^2+{\mathrm{OB}}^2-{\mathrm{OC}}^2=-42$$$${\mathrm{OA}}^2=5,{\mathrm{OB}}^2=13,{\mathrm{OC}}^2=82$$$$\iff {3\mathrm{MO}}^2+3\left(5\right)+13-82=-42$$$$\iff {\mathrm{MO}}^2=12\Rightarrow \mathrm{MO}=\sqrt{12}=2\sqrt{3}$$$$\mathrm{M}\in \mathrm{Circle}\mathrm{center}\mathrm{O}\mathrm{Radius}2\sqrt{3}$$

Commented by mathocean1 last updated on 05/Jan/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}.$$

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