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If-x-6-2-y-1-2-x-9-2-y-5-2-find-minumum-




Question Number 200412 by hardmath last updated on 18/Nov/23
If  (√((x-6)^2  + (y+1)^2 )) + (√((x-9)^2  + (y+5)^2 ))    find:   minumum = ?
$$\mathrm{If} \\ $$$$\sqrt{\left(\mathrm{x}-\mathrm{6}\right)^{\mathrm{2}} \:+\:\left(\mathrm{y}+\mathrm{1}\right)^{\mathrm{2}} }\:+\:\sqrt{\left(\mathrm{x}-\mathrm{9}\right)^{\mathrm{2}} \:+\:\left(\mathrm{y}+\mathrm{5}\right)^{\mathrm{2}} } \\ $$$$ \\ $$$$\mathrm{find}:\:\:\:\mathrm{minumum}\:=\:? \\ $$
Answered by witcher3 last updated on 18/Nov/23
A(6,−1);B=(9,−5)  M(x,y)  we want min AM+MB≥AB;∀M∈IR^2   AM+MB=AB ,∀M∈[AB]  min=(√((6−9)^2 +(−1+5)^2 ))=(√(9+16 ))=5
$$\mathrm{A}\left(\mathrm{6},−\mathrm{1}\right);\mathrm{B}=\left(\mathrm{9},−\mathrm{5}\right) \\ $$$$\mathrm{M}\left(\mathrm{x},\mathrm{y}\right) \\ $$$$\mathrm{we}\:\mathrm{want}\:\mathrm{min}\:\mathrm{AM}+\mathrm{MB}\geqslant\mathrm{AB};\forall\mathrm{M}\in\mathrm{IR}^{\mathrm{2}} \\ $$$$\mathrm{AM}+\mathrm{MB}=\mathrm{AB}\:,\forall\mathrm{M}\in\left[\mathrm{AB}\right] \\ $$$$\mathrm{min}=\sqrt{\left(\mathrm{6}−\mathrm{9}\right)^{\mathrm{2}} +\left(−\mathrm{1}+\mathrm{5}\right)^{\mathrm{2}} }=\sqrt{\mathrm{9}+\mathrm{16}\:}=\mathrm{5} \\ $$
Commented by hardmath last updated on 18/Nov/23
thank you dear professor
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor} \\ $$

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