Question Number 15434 by mrW1 last updated on 10/Jun/17
Commented by mrW1 last updated on 10/Jun/17
$$\mathrm{Two}\:\mathrm{circles}\:\mathrm{with}\:\mathrm{radius}\:\mathrm{6}\:\mathrm{and}\:\mathrm{10}\:\mathrm{and} \\ $$$$\mathrm{center}\:\mathrm{at}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{intersect}\:\mathrm{at}\:\mathrm{point}\:\mathrm{C} \\ $$$$\mathrm{and}\:\mathrm{D}.\:\mathrm{The}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B} \\ $$$$\mathrm{is}\:\mathrm{12}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{max}.\:\mathrm{length}\:\mathrm{of} \\ $$$$\mathrm{segment}\:\mathrm{EF}\:\mathrm{through}\:\mathrm{C}? \\ $$
Commented by ajfour last updated on 10/Jun/17
$$\left({EF}\right)_{{max}} =\mathrm{2}×\left({AB}\right)\:\: \\ $$
Commented by mrW1 last updated on 10/Jun/17
$$\mathrm{please}\:\mathrm{explain}\:\mathrm{your}\:\mathrm{proof}. \\ $$
Commented by ajfour last updated on 10/Jun/17
$${Please}\:{view}\:{Q}.\:\mathrm{15480}\:,\:{Sir}\:. \\ $$
Commented by Tinkutara last updated on 11/Jun/17
$$\mathrm{Sir},\:\mathrm{will}\:{EF}\:\mathrm{be}\:\mathrm{maximum}\:\mathrm{when}\:\mathrm{it}\:\mathrm{is} \\ $$$$\mathrm{parallel}\:\mathrm{to}\:{AB}?\:\mathrm{Because}\:\mathrm{then}\:\mathrm{I}\:\mathrm{can} \\ $$$$\mathrm{prove}\:{EF}\:=\:\mathrm{2}{AB}. \\ $$
Commented by mrW1 last updated on 11/Jun/17
$$\mathrm{Yes},\:\mathrm{EF}\:\mathrm{is}\:\mathrm{max}.\:\mathrm{when}\:\mathrm{EF}\:\mathrm{is}\:\mathrm{parallel} \\ $$$$\mathrm{to}\:\mathrm{AB}. \\ $$$$\mathrm{We}\:\mathrm{can}\:\mathrm{prove}\:\mathrm{this}\:\mathrm{without}\:\mathrm{much} \\ $$$$\mathrm{calculation}: \\ $$$$\mathrm{We}\:\mathrm{look}\:\mathrm{at}\:\mathrm{the}\:\mathrm{triangle}\:\Delta\mathrm{EDF}.\:\mathrm{We} \\ $$$$\mathrm{know}\:\mathrm{no}\:\mathrm{matter}\:\mathrm{where}\:\mathrm{the}\:\mathrm{point}\:\mathrm{E} \\ $$$$\mathrm{and}\:\mathrm{F}\:\mathrm{lie},\:\mathrm{the}\:\mathrm{angle}\:\angle\mathrm{CED}\:\mathrm{and}\:\angle\mathrm{CFD} \\ $$$$\mathrm{remain}\:\mathrm{constant}.\:\mathrm{That}\:\mathrm{means}\:\mathrm{the} \\ $$$$\mathrm{angle}\:\angle\mathrm{EDF}\:\mathrm{is}\:\mathrm{constant}\:\mathrm{when}\:\mathrm{the} \\ $$$$\mathrm{point}\:\mathrm{E}\:\mathrm{and}\:\mathrm{F}\:\mathrm{change}.\:\mathrm{With}\:\mathrm{constant} \\ $$$$\mathrm{angle}\:\angle\mathrm{EDF}\:\mathrm{we}\:\mathrm{know}\:\mathrm{EF}\:\mathrm{is}\:\mathrm{maximum} \\ $$$$\mathrm{when}\:\mathrm{DE}\:\mathrm{and}\:\mathrm{DF}\:\mathrm{are}\:\mathrm{maximum}. \\ $$$$\mathrm{The}\:\mathrm{maximum}\:\mathrm{of}\:\mathrm{DE}\:\mathrm{and}\:\mathrm{DF}\:\mathrm{are} \\ $$$$\mathrm{the}\:\mathrm{diameters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circles}.\:\mathrm{So} \\ $$$$\mathrm{the}\:\mathrm{maximum}\:\mathrm{of}\:\mathrm{EF}\:\mathrm{occurs}\:\mathrm{when} \\ $$$$\mathrm{DE}\:\mathrm{and}\:\mathrm{DF}\:\mathrm{pass}\:\mathrm{through}\:\mathrm{the}\:\mathrm{centers} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{circles},\:\mathrm{in}\:\mathrm{this}\:\mathrm{case}\:\mathrm{EF}\mid\mid\mathrm{AB}\:\mathrm{and} \\ $$$$\mathrm{EF}=\mathrm{2AB}. \\ $$
Commented by mrW1 last updated on 11/Jun/17
Commented by ajfour last updated on 11/Jun/17
$${i}\:{found}\:{a}\:{simpler}\:{way}\:,\:{Sir}. \\ $$$${simpler}\:{to}\:{my}\:{earlier}\:{lengthy} \\ $$$${algebraic}\:{method}.\:{Kindly}\:{view}\:{it} \\ $$$$\left({herein}\:{as}\:{another}\:{answer}\right). \\ $$
Answered by ajfour last updated on 11/Jun/17
Commented by ajfour last updated on 11/Jun/17
$${length}\:{APB}\:=\:{AP}+{PB} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{2}\left({MP}+{PN}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{2}\left({MN}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{2}\left({d}\:\mathrm{cos}\:\theta\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{2}{d}}{\:\sqrt{\mathrm{1}+\mathrm{tan}\:^{\mathrm{2}} \theta}}\: \\ $$$$\:\:{maximum}\:{length}\:{of}\:{APB} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{2}{d}\:\:\:\:\left({for}\:\theta=\mathrm{0}\right)\: \\ $$$$\:\:\:{that}\:{is}\:{when}\:{it}\:{is}\://\:{to}\:{C}_{\mathrm{1}} {C}_{\mathrm{2}} \:. \\ $$
Commented by mrW1 last updated on 11/Jun/17
$$\mathrm{you}\:\mathrm{have}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{Q15480}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{maximum}\:\mathrm{is}\:\mathrm{when}\:\mathrm{m}=\mathrm{0},\:\mathrm{m} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{inclination}\:\mathrm{of}\:\mathrm{line}\:\mathrm{EF}. \\ $$
Commented by ajfour last updated on 11/Jun/17
$${yes},{but}\:{the}\:{same}\:{expression}\:{for} \\ $$$${length}\:{of}\:{EF}\:=\frac{\mathrm{2}{d}}{\:\sqrt{\mathrm{1}+{m}^{\mathrm{2}} }}\:{is}\:{arrived} \\ $$$${at}\:{by}\:{this}\:{method}\:{with}\:{less}\:{effort}. \\ $$
Commented by mrW1 last updated on 11/Jun/17
$$\mathrm{this}\:\mathrm{is}\:\mathrm{better}\:\mathrm{and}\:\mathrm{clear}! \\ $$