Question Number 119007 by bramlexs22 last updated on 21/Oct/20
$${Given}\:{point}\:{A}=\left(\mathrm{7},\mathrm{26}\right)\:{and}\:{B}=\left(\mathrm{12},\mathrm{12}\right)\:,\:{find} \\ $$$${all}\:{points}\:{P}\:{such}\:{that}\:\mid{AP}\:\mid\:=\:\mid\:{BP}\:\mid\:{and}\:\angle{APB}\:=\:\mathrm{90}°\:. \\ $$
Answered by bemath last updated on 21/Oct/20
$$\:{Note}\:{that}\:{the}\:{triangle}\:{ABP}\:{is}\:{an}\:{isosceles}\:{right} \\ $$$${triangle}\:{with}\:\mid{AP}\:\mid\:=\:\mid{BP}\:\mid.\:{Let}\:{Q}\:{be}\:{the}\:{midpoint} \\ $$$${of}\:{segment}\:{AB}.\:{Then}\:{Q}=\left(\frac{\mathrm{19}}{\mathrm{2}}\:,\mathrm{19}\right),\:{where} \\ $$$$\mid{QA}\mid\:=\:\mid{QB}\mid\:=\:\mid{QP}\:\mid\:\:{and}\:{QA}\:\bot\:{QB} \\ $$$${Thus}\:\overset{\rightarrow} {{Q}}\overset{\rightarrow} {{A}}\:=\:\left(\frac{\mathrm{5}}{\mathrm{2}},−\mathrm{7}\right)\:{and}\:\overset{\rightarrow} {{Q}P}\:=\:\left(\mathrm{7},\frac{\mathrm{5}}{\mathrm{2}}\right)\: \\ $$$${or}\:\left(−\mathrm{7},−\frac{\mathrm{5}}{\mathrm{2}}\right).\:{Let}\:{O}=\left(\mathrm{0},\mathrm{0}\right)\:{be}\:{the}\:{origin}\:,\:{it}\:{follows}\:{that} \\ $$$${O}\overset{\rightarrow} {{P}}\:=\:{O}\overset{\rightarrow} {{Q}}\:+\:{Q}\overset{\rightarrow} {{P}}\:=\:\left(\frac{\mathrm{19}}{\mathrm{2}},\:\mathrm{19}\right)\:\pm\left(\mathrm{7},\frac{\mathrm{5}}{\mathrm{2}}\right). \\ $$$${Consequently}\:{P}\:=\:\left(\frac{\mathrm{33}}{\mathrm{2}},\:\frac{\mathrm{43}}{\mathrm{2}}\right)\:{or}\:\left(\frac{\mathrm{5}}{\mathrm{2}},\:\frac{\mathrm{33}}{\mathrm{2}}\right) \\ $$
Answered by 1549442205PVT last updated on 21/Oct/20