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

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),(C;−1)} Question 1 knowing that 3MA^2 +MB^2 −MC^2 =3MO^2 +3OA^2 +OB^2 −OC^2 Determine and construct the set of points M on the plane such as 3MA^2 +MB^2 −MC^2 =−42

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

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

3MA^2 +MB^2 −MC^2 =3MO^2 +3OA^2 +OB^2 −OC^2 =−42 OA^2 =5,OB^2 =13,OC^2 =82 ⇔3MO^2 +3(5)+13−82=−42 ⇔MO^2 =12⇒MO=(√(12))=2(√3) M∈Circle center O Radius 2(√3)

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

Commented by mathocean1 last updated on 05/Jan/20

thank you sir.

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

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