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UMA ABORDAGEM DO ESTUDO DE CÔNICAS E QUÁDRICAS COM O AUXÍLIO DO SOFTWARE GEOGEBRA

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❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❈â♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦

❯♠❛ ❛❜♦r❞❛❣❡♠ ❞♦ ❡st✉❞♦ ❞❡ ❝ô♥✐❝❛s ❡

q✉á❞r✐❝❛s ❝♦♠ ♦ ❛✉①í❧✐♦ ❞♦ s♦❢t✇❛r❡

●❡♦●❡❜r❛

▲✉✐③ ❋❡r♥❛♥❞♦ ●✐♦❧♦ ❆❧✈❡s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r✲ ❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡

❖r✐❡♥t❛❞♦r❛

Pr♦❢❛✳ ❉r❛✳ ❆❧✐❝❡ ❑✐♠✐❡ ▼✐✇❛ ▲✐❜❛r❞✐

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Alves, Luiz Fernando Giolo

Uma abordagem do estudo de cônicas e qurdricas com o auxílio do software GeoGebra / Luiz Fernando Giolo Alves. - Rio Claro, 2016 79 f. : il., figs., grrfs., tabs.

Dissertação (mestrado) - Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas

Orientador: Alice Kimie Miwa Libardi

1. Geometria analítica. 2. Qurdricas. 3. Cônicas. 4. GeoGebra. I. Título.

516.3 A474a

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❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

▲✉✐③ ❋❡r♥❛♥❞♦ ●✐♦❧♦ ❆❧✈❡s

❯♠❛ ❛❜♦r❞❛❣❡♠ ❞♦ ❡st✉❞♦ ❞❡ ❝ô♥✐❝❛s ❡ q✉á❞r✐❝❛s ❝♦♠ ♦

❛✉①í❧✐♦ ❞♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛

❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

Pr♦❢❛✳ ❉r❛✳ ❆❧✐❝❡ ❑✐♠✐❡ ▼✐✇❛ ▲✐❜❛r❞✐ ❖r✐❡♥t❛❞♦r❛

Pr♦❢✳ ❉r✳ ❚❤✐❛❣♦ ❞❡ ▼❡❧♦ ■●❈❊✴❯◆❊❙P✴❘✐♦ ❈❧❛r♦ ✭❙P✮

Pr♦❢❛✳ ❉r❛✳ ❚❛t✐❛♥❛ ▼✐❣✉❡❧ ❘♦❞r✐❣✉❡s ❋❈✴❯◆❊❙P✴❇❛✉r✉ ✭❙P✮

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♠ã❡ ❘❡❣✐♥❛ ♣❡❧♦s ❡①❡♠♣❧♦s ❞❡ ✈✐❞❛ ❡ ♣❡❧❛s ♦♣♦rt✉♥✐❞❛❞❡s q✉❡ ❢♦r♥❡❝❡✉✱ ♣♦✐s ❢♦r❛♠ ✐♠♣r❡s❝✐♥❞í✈❡✐s ♣❛r❛ q✉❡ ❡✉ ❝❤❡❣❛ss❡ ❛té ❡st❡ ♠♦♠❡♥t♦✳

➚ ♠✐♥❤❛ ♥♦✐✈❛ ❇r✉♥❛ ♣❡❧♦ ❛✉①í❧✐♦ ❡ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ♥♦s ✏❞r❛♠❛s✑ ❛❝❛❞ê♠✐❝♦s q✉❡ ❝♦♠♣❛rt✐❧❤❛♠♦s✳

➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ❆❧✐❝❡ ♣❡❧❛ ❛❥✉❞❛ ❡ ❡♥♦r♠❡ ❝♦♠♣r❡❡♥sã♦✱ ❡ss❡♥❝✐❛✐s ♣❛r❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

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❘❡s✉♠♦

❖ t❡①t♦ q✉❡ s❡❣✉❡ ❛❜♦r❞❛ ✉♠ ❡st✉❞♦ ❞❡ ❝ô♥✐❝❛s ❡ q✉á❞r✐❝❛s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❛✉①✐❧✐❛r ♣r♦❢❡ss♦r❡s ❡ ❡st✉❞❛♥t❡s ❛ t❡r ✉♠❛ ✈✐sã♦ ♠❛✐s ❝♦♥❝r❡t❛ ❡ ❞✐♥â♠✐❝❛ ❞❡st❡s ❡❧❡✲ ♠❡♥t♦s ❝♦♠ ♦ s♦❢t✇❛r❡ ❞❡ ❞✐str✐❜✉✐çã♦ ❧✐✈r❡ ●❡♦●❡❜r❛✳ ◆✉♠ ♣r✐♠❡✐r♦ ♠♦♠❡♥t♦ t❡♠♦s ❝♦♠♦ ❛❧✈♦ ♦❜s❡r✈❛çõ❡s s♦❜r❡ ❝ô♥✐❝❛s ❝♦♠ ❞✐❝❛s ❞❡ ❝♦♠♦ ❞✐r✐❣✐r✲s❡ ❛♦ ❛ss✉♥t♦ ✉s❛♥❞♦ ❛s ❢❡rr❛♠❡♥t❛s q✉❡ ♦ ●❡♦●❡❜r❛ tr❛③ ♣❛r❛ ❢❛❝✐❧✐t❛r ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦s s✐❣♥✐✜❝❛❞♦s ❞♦s ♣❛✲ râ♠❡tr♦s ❡ ❝♦❡✜❝✐❡♥t❡s ❞❡ss❛s ❡q✉❛çõ❡s q✉❛❞rát✐❝❛s✳ ❊♠ s❡❣✉✐❞❛✱ ❡st✉❞❛♠♦s ❛❧❣✉♠❛s ♣❛rt✐❝✉❧❛r✐❞❛❞❡s ❞❛s q✉á❞r✐❝❛s✱ ❛ss✉♥t♦ q✉❡ ✉s✉❛❧♠❡♥t❡ ♥ã♦ é ✈✐st♦ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳

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❆❜str❛❝t

❚❤❡ ❢♦❧❧♦✇✐♥❣ t❡①t ✐s ❛ st✉❞② ♦❢ ❝♦♥✐❝s ❛♥❞ q✉❛❞r✐❝s ✇✐t❤ ❛ ❣♦❛❧ t♦ ❣✐✈❡ ❛ ❝♦♥❝r❡t❡ ❛♥❞ ❞✐♥❛♠✐❝ ❛♣♣r♦❛❝❤ ♦❢ t❤❡ s✉❜❥❡❝t ✇✐t❤ t❤❡ ❛ss✐st❛♥❝❡ ♦❢ t❤❡ ❢r❡❡ s♦❢t✇❛r❡ ●❡♦●❡❜r❛✳ ❆t ✜rst ♠♦♠❡♥t ✇❡ t❛r❣❡t s♦♠❡ ♦❜s❡r✈❛t✐♦♥s ❛❜♦✉t ❝♦♥✐❝s✱ ✇✐t❤ t✐♣s ♦❢ ❤♦✇ t♦ ❛♣♣r♦❛❝❤ t❤❡ s✉❜❥❡❝t ✇✐t❤ t❤❡ t♦♦❧s t❤❛t ●❡♦●❡❜r❛ ❜r✐♥❣s t♦ ♠❛❦❡ ❡❛s✐❡r t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ♠❡❛♥✐♥❣s ♦❢ t❤❡ ♣❛r❛♠❡t❡rs ❛♥❞ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡s❡ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✳ ❆♥❞ t❤❡♥ ✇❡ st✉❞② s♦♠❡ ♣❛rt✐❝✉❧❛r✐t✐❡s ♦❢ t❤❡ q✉❛❞r✐❝ s✉r❢❛❝❡s✱ ❛ s✉❜❥❡❝t t❤❛t ✉s✉❛❧❧② ✐s ♥♦t s❡❡♥ ✐♥ t❤❡ ❤✐❣❤ s❝❤♦♦❧✳

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▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❏❛♥❡❧❛ ♣❛❞rã♦ ❞♦ ●❡♦●❡❜r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷ ▼❡♥✉ ❡①✐❜✐r✱ ❢✉♥❝✐♦♥❛❧✐❞❛❞❡s ❡ ❛t❛❧❤♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✸ ❚❡❝❧❛❞♦s ✈✐rt✉❛✐s ❡♥❝♦♥tr❛❞♦s ❡♠ ❊①✐❜✐r ❚❡❝❧❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✶✳✹ ❏❛♥❡❧❛ ❞❡ ➪❧❣❡❜r❛ ❝♦♠ ❛❧❣✉♥s ❡❧❡♠❡♥t♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✺ ▼❡♥✉ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❏❛♥❡❧❛ ❞❡ ❱✐s✉❛❧✐③❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✶✳✻ ❈❛r❛❝t❡r❡s ❡s♣❡❝✐❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✶ P❛râ♠❡tr♦ ❝r✐❛❞♦✳ P♦❞❡♠♦s ♥♦♠❡á✲❧♦ ❝♦♠♦ ❞❡s❡❥❛r♠♦s✳ a✱ k✱ b✱cte✱ ❡t❝✳ ✷✻ ✷✳✷ ❊✐s ♦ ♣❧❛♥♦ ♥❛ ❛❧t✉r❛z ❡♠ q✉❡ a ❢♦✐ s❡t❛❞♦✱ ♥♦ ❝❛s♦ a= 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✷✳✸ ❊s♣❛ç♦ ❝♦♠ ♦s ❡❧❡♠❡♥t♦s ❝r✐❛❞♦s✱ ✉♠ ❝♦♥❡ ❡ ✉♠ ♣❧❛♥♦ q✉❡ ♦ ✐♥t❡rs❡❝t❛✳ ✷✼ ✷✳✹ ❇♦tã♦ ♣❛r❛ ❣✐r♦ ❞♦ ❡s♣❛ç♦ ❡♠ t❡❧❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✺ ❆t✐✈❛♥❞♦ ✈❛r✐❛çã♦ ❛✉t♦♠át✐❝❛ ❞♦ ♣❛râ♠❡tr♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✻ ❇♦tã♦ ♣❛r❛ ❣❡r❛r ✐♥t❡rs❡❝çã♦ ❡♥tr❡ s✉♣❡r❢í❝✐❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✼ ■♥t❡rs❡❝çã♦ ❡♥tr❡ ♦ ❝♦♥❡ ❡ ♦ ♣❧❛♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✽ P❛r❛ ♦❝✉❧t❛r ♦ ♦❜❥❡t♦ ❜❛st❛ ❞❡s♠❛r❝❛r ❛ ❧✐♥❤❛ ❊①✐❜✐r ♦❜❥❡t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✷✳✾ ❊①✐❜✐♥❞♦ ❛♣❡♥❛s ♦ r❡s✉❧t❛❞♦ ❞❛ ✐♥t❡rs❡❝çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✶✵ ❖❜❥❡t♦s ♦❝✉❧t♦s ♥❛ ❏❛♥❡❧❛ ❞❡ ❱✐s✉❛❧✐③❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✷✳✶✶ ❈♦♥tr♦❧❡s ❞❡s❧✐③❛♥t❡s ♣❛r❛ ❛❧t❡r❛çã♦ ❞❡ ✈❛❧♦r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✶✷ ❊q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ s❡✉ r❡s✉❧t❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✶✸ ❏❛♥❡❧❛ ♦♥❞❡ ❝♦♥✜❣✉r❛♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♦❜❥❡t♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✶✹ ❈♦♦r❞❡♥❛❞❛s s❡♥❞♦ ❡①✐❜✐❞❛s ❝♦♠♦ rót✉❧♦ ❞♦ ♣♦♥t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✶✺ ▼✉❞❛♥❞♦ ♦ ❢♦r♠❛t♦ ❞❡ ❡①✐❜✐çã♦ ❞❛ ❡q✉❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✶✻ ■♥t❡rs❡❝çã♦ ❡♥tr❡ ♦ ❝♦♥❡ ❡ ♦ ♣❧❛♥♦ ❣❡r❛♥❞♦ ✉♠❛ ❡❧✐♣s❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✶✼ ❈r✐❛♥❞♦ ✈✐st❛ ✷❉ ❞❛ ❡❧✐♣s❡ ❣❡r❛❞❛ ♣❡❧❛ ✐♥t❡rs❡❝çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✶✽ ❊❧✐♣s❡ (x−m)2

a2 +

(y−n)2

b2 = 1 ❝♦♠ a= 2 ❡ b=m=n = 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✷✳✶✾ ❇♦tã♦ q✉❡ ❣❡r❛ ♣♦♥t♦s ♥♦ ♣❧❛♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✷✵ ❊s❝♦❧❤❡♥❞♦ ❛ ❢✉♥çã♦ s❡❣♠❡♥t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✷✶ ■♥t❡rs❡❝çã♦ ❡♥tr❡ ♦ ❝♦♥❡ ❡ ♦ ♣❧❛♥♦ ❣❡r❛♥❞♦ ✉♠❛ ♣❛rá❜♦❧❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✷✷ ❈r✐❛♥❞♦ ❛♥✐♠❛çã♦ ❞♦ ♣❧❛♥♦ ✐♥t❡rs❡❝t❛♥❞♦ ♦ ❝♦♥❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✷✸ P❛rá❜♦❧❛ ❞❡ ❡q✉❛çã♦ (yb)2 = 2p(xa) ❝♦♠ a=b= 1 p= 2✳ ✳ ✳ ✳ ✹✷

(14)

✷✳✷✻ ❊s❝♦♥❞✐❞♦s ♦s ❡❧❡♠❡♥t♦s ❛✉①✐❧✐❛r❡s ❞❛ ❝♦♥str✉çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✷✼ ❉❛ ♠❛♥❡✐r❛ q✉❡ r❡❢❡r❡♥❝✐❛♠♦s✱ ♦ ❢♦❝♦ ❡ ❛ ❞✐r❡tr✐③ s❡❣✉❡♠ ❛ ♣❛rá❜♦❧❛✳ ✳ ✹✻ ✷✳✷✽ ❊♠ ❞❡st❛q✉❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ C ❛♦ ❢♦❝♦A ❡ à ❞✐r❡tr✐③ h✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✷✾ ❉✐r❡tr✐③ ❡ ❢♦❝♦ q✉❡ s❡rã♦ ♣❛râ♠❡tr♦s ♣❛r❛ ❛ ♣❛rá❜♦❧❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✸✵ g é ❛ ▼❡❞✐❛tr✐③ ❞♦ s❡❣♠❡♥t♦CD✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✸✶ ■♥t❡rs❡❝çã♦ ❡♥tr❡ ❛s r❡st❛s g ❡ h✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✸✷ ❍❛❜✐❧✐t❛r ♦ r❛str♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✷✳✸✸ P❛rá❜♦❧❛ ❣❡r❛❞❛ ♣❡❧♦ r❛str♦ ❞♦ ♣♦♥t♦ E✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✷✳✸✹ ■♥t❡rs❡❝çã♦ ❡♥tr❡ ♦ ❝♦♥❡ ❡ ♦ ♣❧❛♥♦ ❣❡r❛♥❞♦ ✉♠❛ ❤✐♣ér❜♦❧❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✷✳✸✺ ❱✐st❛ ✷❉ ❞❛ ❤✐♣ér❜♦❧❡ ❣❡r❛❞❛ ♣❡❧❛ ✐♥t❡rs❡❝çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✸✻ ❍✐♣ér❜♦❧❡ (x−m)2

a2 −

(y−n)2

b2 = 1 ❝♦♠ a = 2 ❡b =m=n= 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✷✳✸✼ ❍✐♣ér❜♦❧❡ (x−m)2

a2 −

(y−n)2

b2 = 1✱ ❝♦♠ a= 2 ❡b =m =n = 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✷✳✸✽ ❱❛❧♦r ❛❜s♦❧✉t♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ AF1 ❡AF2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✸✳✶ ❆❜r✐♥❞♦ ❛ ❥❛♥❡❧❛ ❞❛ ❝❛❧❝✉❧❛❞♦r❛ ❛❧❣é❜r✐❝❛ ❞♦ ●❡♦●❡❜r❛ ❏❛♥❡❧❛ ❈❆❙✳ ✳ ✳ ✺✼

✸✳✷ ❆♠❜✐❡♥t❡ ❞♦ ●❡♦●❡❜r❛ ❞❡♣♦✐s q✉❡ ❞❡s❝♦❜r✐♠♦s ♦s ❛✉t♦✈❛❧♦r❡s ❡ ❛✉t♦✲ ✈❡t♦r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✸ ▼✉❞❛♥❞♦ ❛ ❡①✐❜✐çã♦ ♣❛r❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✹ ❊❧✐♣s❡ ❝♦♠ s❡✉ ❡✐①♦ r♦t❛❝✐♦♥❛❞♦ ❡ tr❛♥s❧❛❞❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✳✶ Pr✐♥❝✐♣❛✐s q✉á❞r✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✹✳✷ ❙❡ ❞✐❣✐t❛❞♦ ♥❛ ❡♥tr❛❞❛ xy = 1 ❝♦♠ ❏❛♥❡❧❛ ❞❡ ❱✐s✉❛❧✐③❛çã♦ ❡ ❛ ❏❛♥❡❧❛ ✸❉

(15)

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶✺

✶✳✶ ❆❧❣✉♥s ❞❛❞♦s ❤✐stór✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷ ●❡♦●❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✶ ❇❛rr❛ ❞❡ ♠❡♥✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷✳✷ ❇❛rr❛ ❞❡ ❢❡rr❛♠❡♥t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✷✳✸ ❏❛♥❡❧❛ ❞❡ á❧❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✷✳✹ ❏❛♥❡❧❛ ❞❡ ✈✐s✉❛❧✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✷✳✺ ❈❛♠♣♦ ❞❡ ❡♥tr❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✷ ❈ô♥✐❝❛s ✷✺

✷✳✶ ❈✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✶✳✶ ❙❡çã♦ ❝ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✶✳✷ ❊q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✷✳✶ ❙❡çã♦ ❝ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✷✳✷ ❊q✉❛çã♦ ❞❛ ❡❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✸ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✸✳✶ ❙❡çã♦ ❝ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✸✳✷ ❊q✉❛çã♦ ❞❛ ♣❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✸✳✸ ●❡r❛♥❞♦ ❛ ♣❛rá❜♦❧❛ ❛ ♣❛rt✐r ❞❛ ❞❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✹ ❍✐♣ér❜♦❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✷✳✹✳✶ ❙❡çã♦ ❝ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✷✳✹✳✷ ❊q✉❛çã♦ ❞❛ ❤✐♣ér❜♦❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✸ ■❞❡♥t✐✜❝❛çã♦ ❞❡ ❝ô♥✐❝❛s ✺✼

✸✳✵✳✶ ❈á❧❝✉❧♦ ❞♦s ❛✉t♦✈❛❧♦r❡s ❡ ❛✉t♦✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

✹ ❆❧❣✉♠❛s ♦❜s❡r✈❛çõ❡s s♦❜r❡ q✉á❞r✐❝❛s ✻✸

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✺ ❘♦t❛çã♦✱ tr❛♥s❧❛çã♦✱ ✐♥✈❛r✐❛♥t❡s ❡ ❝❡♥tr♦ ✻✼ ✺✳✶ ❉❡✜♥✐♥❞♦ ✉♠ ♥♦✈♦ s✐st❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✺✳✷ ❈❡♥tr♦ ❞❛s q✉á❞r✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✺✳✷✳✶ ❆♥❛❧✐s❛♥❞♦ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♦ ❝❡♥tr♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

✻ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✼✼

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✶ ■♥tr♦❞✉çã♦

✶✳✶ ❆❧❣✉♥s ❞❛❞♦s ❤✐stór✐❝♦s

❉✐③ ❛ ❧❡♥❞❛ q✉❡ ✉♠❛ ♣❡st❡ ❛ss♦❧❛✈❛ ❆t❡♥❛s ♥♦ sé❝✉❧♦ ■❱ ❛✳❈✳ ❡ ♦s ❣r❡❣♦s ❛♦ ❝♦♥s✉❧✲ t❛r❡♠ ♦ ♦rá❝✉❧♦ ❞❡ ❉❡❧♦s ❢♦r❛♠ ❛❝♦♥s❡❧❤❛❞♦s ❛ ❞✉♣❧✐❝❛r❡♠ ♦ ❛❧t❛r ❞❡ ❆♣♦❧♦ ♣❛r❛ q✉❡ ❛ ♣❡st❡ ❢♦ss❡ ❛♠❡♥✐③❛❞❛✳

❊ss❡ ❛❧t❛r t✐♥❤❛ ✉♠❛ ❢♦r♠❛ ❝ú❜✐❝❛✱ ❡♥tã♦ ♥ã♦ s❡r✐❛ ♠✉✐t♦ tr❛❜❛❧❤♦s♦ ❢❛③❡r ❛ ♦❜r❛ ❞❡ ❛♠♣❧✐❛çã♦✳ ❖ ♣❡ss♦❛❧ ♥ã♦ t❡✈❡ ❞ú✈✐❞❛s ❡ ❧♦❣♦ ❞✉♣❧✐❝❛r❛♠ t♦❞❛s ❛s ♠❡❞✐❞❛s ❞♦ ❧♦❝❛❧✱ ♣♦ré♠ ♣❛r❡❝❡ q✉❡ ❛ ♦❜r❛ t❡✈❡ ✉♠ ❡❢❡✐t♦ ❝♦♥trár✐♦ ❛♦ ❞❡s❡❥❛❞♦ ❡ ❛ ♣❡st❡ ♥❛ ✈❡r❞❛❞❡ s❡ ✐♥t❡♥s✐✜❝♦✉✱ ♣♦rt❛♥t♦ ♦s ❝♦♥str✉t♦r❡s r❡s♦❧✈❡r❛♠ ♣r♦❝✉r❛r ♦s ♠❛t❡♠át✐❝♦s ❞❛ ❆❝❛❞❡♠✐❛ ❞❡ P❧❛tã♦ ♣❛r❛ q✉❡ r❡✈✐s❛ss❡♠ ♦ ♣r♦❥❡t♦✳

▲á ❢♦✐ ❝♦♥st❛t❛❞♦ q✉❡ ❛♦ ❞✉♣❧✐❝❛r ❛s ❛r❡st❛s ❞❡ss❡ ❝✉❜♦ s❡✉ ✈♦❧✉♠❡ ♥❛ ✈❡r❞❛❞❡ ❢♦✐ ♦❝t✉♣❧✐❝❛❞♦ ❡ ♦ t❡♠♣❧♦ ✜❝♦✉ ✹ ✈❡③❡s ♠❛✐♦r ❞♦ q✉❡ r❡❛❧♠❡♥t❡ ❞❡✈❡r✐❛✳

➱ ❝❧❛r♦ q✉❡ s❛❜❡♠♦s q✉❡ ❛ ♣❡st❡ s❡ ✐♥t❡♥s✐✜❝♦✉ ♣♦rq✉❡ ♦ ♣❡ss♦❛❧ r❡s♦❧✈❡✉ ❛♠♣❧✐❛r ♦ t❡♠♣❧♦ ❛♦ ✐♥✈és ❞❡ ❢❛③❡r ❛s ♣r♦✜❧❛①✐❛s ♣❛r❛ tr❛t❛r ❡ ❡✈✐t❛r ♦ ❛❧❛str❛♠❡♥t♦ ❞❛ ❞♦❡♥ç❛✱ ♠❛s ❤á ❞❡ s❡ ♥♦t❛r q✉❡ s✉r❣✐✉ ✉♠ ❞❡s❛✜♦ ♠✉✐t♦ ✐♥t❡r❡ss❛♥t❡ q✉❡ ❢♦✐ ♦ ❢❛♠♦s♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✉♣❧✐❝❛çã♦ ❞♦ ❝✉❜♦✳

▼❡♥❛❡❝♠✉s ❡r❛ ❞✐s❝í♣✉❧♦ ❞❡ ❊✉❞ó①✐♦ ❡ ♠❡♠❜r♦ ❞❛ ❆❝❛❞❡♠✐❛ ❞❡ P❧❛tã♦✱ ❡❧❡ ❢♦✐ ✉♠ ❞♦s ♠❛t❡♠át✐❝♦s q✉❡ s❡ ♣r♦♣ôs ❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✉♣❧✐❝❛çã♦ ❞♦ ❝✉❜♦ ❡ s✉❛ ❛❜♦r❞❛❣❡♠ s♦❜r❡ ♦ ❛ss✉♥t♦ ♦ ❧❡✈♦✉ ❛ ❞❡s❝♦❜r✐r ❛s s❡çõ❡s ❝ô♥✐❝❛s✿ ❡❧✐♣s❡✱ ♣❛rá❜♦❧❛ ❡ ❤✐♣ér❜♦❧❡✳

❱❛❧❡ ❧❡♠❜r❛r q✉❡ ❛ss✐♠ ❝♦♠♦ ♦s r❡s♣♦♥sá✈❡✐s ♣❡❧❛ s❛ú❞❡ ♥ã♦ t✐♥❤❛♠ ♦s ♣r♦t♦❝♦❧♦s✱ r❡♠é❞✐♦s ❡ ❢❡rr❛♠❡♥t❛s q✉❡ t❡♠♦s ❛t✉❛❧♠❡♥t❡✱ ▼❡♥❛❡❝♠✉s t❛♠❜é♠ ♥ã♦ ♣♦ss✉í❛ ❛s ♥♦t❛çõ❡s ♠♦❞❡r♥❛s q✉❡ ❢❛❝✐❧✐t❛r✐❛♠ ✐♠❡♥s❛♠❡♥t❡ s✉❛s ❞❡❞✉çõ❡s✳ ❚❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ ❛ ❛❜♦r❞❛❣❡♠ ❞❛ ♠❛t❡♠át✐❝❛ ♥❛ ●ré❝✐❛ ❛♥t✐❣❛ ❡r❛ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❣❡♦♠étr✐❝❛✱ ♦ ♣r♦❜❧❡♠❛ s❡ t♦r♥♦✉ ♠❛✐s ❞❡s❛✜❛❞♦r✱ ❛✐♥❞❛ ♠❛✐s s❡ ❧❡✈❛r♠♦s ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦ ❢❛t♦ ❞❡s❝♦❜❡rt♦ ❡♠ ✶✽✸✼✱ ♣♦✐s ♦ ❢r❛♥❝ês P✐❡rr❡ ▲✳ ❲❛♥t❣❡❧ ❞❡♠♦♥str♦✉ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡❧✐❛♥♦ ✭❝♦♠♦ é ❝❤❛♠❛❞♦ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✉♣❧✐❝❛çã♦✮ ♥ã♦ ❛❞♠✐t❡ s♦❧✉çã♦ ❝♦♠ ✉s♦ ❞❡ ré❣✉❛ ❡ ❝♦♠♣❛ss♦ ❛♣❡♥❛s✳

P♦✐s ♠❡s♠♦ ❛ss✐♠ ▼❡♥❛❡❝♠✉s s♦❧✉❝✐♦♥♦✉ ♦ ♣r♦❜❧❡♠❛ ❡ s✉❛ ❛❜♦r❞❛❣❡♠ ❝✉❧♠✐♥♦✉ ♥❛ ❞❡s❝♦❜❡rt❛ ❞❛s s❡çõ❡s ❝ô♥✐❝❛s✱ ♣♦ré♠ ♦s ♥♦♠❡s ❡❧✐♣s❡✱ ♣❛rá❜♦❧❛ ❡ ❤✐♣ér❜♦❧❡ ✈✐r✐❛♠ ♣♦st❡r✐♦r♠❡♥t❡ ❝♦♠ ❆♣♦❧ô♥✐♦✱ ♠❛t❡♠át✐❝♦ ❞❛ ❝♦❧ô♥✐❛ ❣r❡❣❛ P❡r❣❛✳ ❊❧❡ ❜✉s❝♦✉ ♦s

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✶✻ ■♥tr♦❞✉çã♦

♥♦♠❡s ♥❛ ♠❛t❡♠át✐❝❛ ❞❛ ❡s❝♦❧❛ ♣✐t❛❣ór✐❝❛ ♥❛ ❛♣❧✐❝❛çã♦ ♥♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ s✉❛s três ❢♦r♠❛s✿ ♣♦r ❢❛❧t❛✱ s✐♠♣❧❡s ❡ ♣♦r ❡①❝❡ss♦ q✉❡ sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ❡❧✐♣s❡✱ ♣❛rá❜♦❧❛ ❡ ❛ ❤✐♣ér❜♦❧❡✳

❏á ❛s q✉á❞r✐❝❛s✱ ✈ê♠ ❞❡ ❆❧❡①❛♥❞r✐❛ ♣♦r ❞♦✐s tr❛t❛❞♦s ❞♦ ❣❡♥✐❛❧ ❆rq✉✐♠❡❞❡s q✉❡ ❛❜♦r❞❛♠ ❞❡ ❢♦r♠❛ ❛♣r♦❢✉♥❞❛❞❛ só❧✐❞♦s ❞❡ r❡✈♦❧✉çã♦✳ ❙ã♦ ❡❧❡s ✏❙♦❜r❡ ❝♦♥ó✐❞❡s ❡ ❡s❢❡✲ ró✐❞❡s✑ ❡ ✏❙♦❜r❡ ❊s❢❡r❛s ❡ ❈✐❧✐♥❞r♦s✑✱ ♥❡❧❡s ❆rq✉✐♠❡❞❡s ❞❡s❝r❡✈❡ só❧✐❞♦s ❞❡ r❡✈♦❧✉çã♦ ❢♦r♠❛❞♦s ♣♦r ❡❧✐♣s❡s✱ ♣❛rá❜♦❧❛s ❡ ❤✐♣ér❜♦❧❡s r♦t❛❝✐♦♥❛❞♦s ❡♠ t♦r♥♦ ❞❡ s❡✉s ❡✐①♦s ❡ t❛♠❜é♠ ❞❡♠♦♥str❛çõ❡s ❞❛s ár❡❛s ❞❡ ❡❧✐♣s❡s✱ ár❡❛s ❡ ✈♦❧✉♠❡s ❞❡ ❡s❢❡r❛s✱ ❝✐❧✐♥❞r♦s ❡ ❞❛s s✉♣❡r✜❝✐❡s ✱♦❜t✐❞❛s ♣♦r s✉❛s s❡❝çõ❡s✳ ❊♠ ✶✻✷✾✱ P✐❡rr❡ ❞❡ ❋❡r♠❛t ❝♦♥❝❧✉✐ ♦ ♠❛♥✉s❝r✐t♦ ❞❡ ✏■♥tr♦❞✉çã♦ ❛♦s ❧✉❣❛r❡s ♣❧❛♥♦s ❡ só❧✐❞♦s✑ q✉❡ tr❛③ ✉♠❛ ♣❡r❝❡♣çã♦ ❞❡ ✉♠❛ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛ ❡♠ três ❞✐♠❡♥sõ❡s ♦♥❞❡ é ❞✐t♦ ✏❙❡ ♦ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦ ❡♥✈♦❧✈❡ três ✐♥❝ó❣♥✐✲ t❛s✱ ❞❡✈❡✲s❡ ❛❝❤❛r✱ ♣❛r❛ s❛t✐s❢❛③❡r ❛ ❡q✉❛çã♦✱ ♥ã♦ ❛♣❡♥❛s ✉♠ ♣♦♥t♦ ♦✉ ✉♠❛ ❝✉r✈❛✱ ♠❛s t♦❞❛ ✉♠❛ s✉♣❡r❢í❝✐❡✑✳

◆♦ ❡♥t❛♥t♦ ❞❡✈❡✲s❡ ❛ ❊✉❧❡r ✉♠❛ ❞❛s ❣r❛♥❞❡s ❝♦♥tr✐❜✉✐çõ❡s ❞❛ ❣❡♦♠❡tr✐❛ ♥♦R3 ✱ ♣♦✐s

❡❧❡ ❛♣r❡s❡♥t❛ ❡♠ s❡✉ ❧✐✈r♦ ✏■♥tr♦❞✉çã♦ à ❆♥á❧✐s❡ ■♥✜♥✐t❛✑✱ ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✼✹✽ ❛ ♣r✐♠❡✐r❛ ❡①♣♦s✐çã♦ ❡♠ ❧✐✈r♦✲t❡①t♦ ❞❡ s✉♣❡r❢í❝✐❡s q✉á❞r✐❝❛s ❝♦♥s✐❞❡r❛♥❞♦✲❛s ❝♦♠♦ s✉♣❡r❢í❝✐❡s ❞❡ ❣r❛✉ ✷ ♥♦ R3✳

❊ ❣r❛ç❛s ❛ t♦❞❛ ❡ss❛ ♣❡sq✉✐s❛ ❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❛♦ ❧♦♥❣♦ ❞♦s sé❝✉❧♦s✱ ♣♦❞❡♠♦s ❤♦❥❡ ❡♥s✐♥❛r t❛❧ ❝♦♥t❡ú❞♦ ❡ r❛❝✐♦❝í♥✐♦ ❛ ❥♦✈❡♥s ❡♠ ✐❞❛❞❡ ❡s❝♦❧❛r✳

◆❡st❛ ❞✐ss❡rt❛çã♦ ✐r❡♠♦s s✉❣❡r✐r ✉♠❛ ❛❜♦r❞❛❣❡♠ ✈✐s✉❛❧ ❡ ❞✐♥â♠✐❝❛ ❝♦♠ ♦ ❛✉①í✲ ❧✐♦ ❞♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛ ♣❛r❛ ❝♦♠♣❧❡♠❡♥t❛r ❛s ❛✉❧❛s ❞♦ ♣r♦❢❡ss♦r ❞❡ ❡♥s✐♥♦ ❜ás✐❝♦ ♦✉ s✉♣❡r✐♦r✱ ♣♦✐s tr❛t❛r❡♠♦s ❞❛s ❝ô♥✐❝❛s✱ s✉❛s ❡q✉❛çõ❡s r❡❞✉③✐❞❛s✱ ❛♣r❡s❡♥t❛çã♦ ❡ ♦ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ✉♠❛ ❝ô♥✐❝❛ ❣❡r❛❧✳

❆❜♦r❞❛r❡♠♦s t❛♠❜é♠ ❛s ❢♦r♠❛s q✉á❞r✐❝❛s✱ ❡st❛s ♣♦r ♥ã♦ ❡st❛r❡♠ ♥❛ ❣r❛❞❡ ❝✉r✲ r✐❝✉❧❛r ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ t❡rã♦ ✉♠ ❝❛♣ít✉❧♦ ❞❡ ❞✐s❝✉ssã♦ ❞❡ s✉❛ ❡q✉❛çã♦ ❣❡r❛❧✱ ❢♦r♠❛ ♠❛tr✐❝✐❛❧ ❡ tr❛♥s❢♦r♠❛çõ❡s ❞❡ r♦t❛çã♦ ❡ tr❛♥s❧❛çã♦✳ ❉❡♣♦✐s ❞❡ ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ✈✐s✉❛❧ ❞❡ ❝❛❞❛ ✉♠❛ ❞❡ s✉❛s ❢♦r♠❛s ❝♦♠ ♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛✳

❈♦♠ ❡st❡ tr❛❜❛❧❤♦ ❡s♣❡r❛♠♦s ❡♥tr❡❣❛r ✉♠ ♠❛t❡r✐❛❧ q✉❡ ❛✉①✐❧✐❡ ❡st✉❞❛♥t❡s ❡ ♣r♦✲ ❢❡ss♦r❡s ♥❛ ❛♣r❡s❡♥t❛çã♦ ❡ ❛♣r♦❢✉♥❞❛♠❡♥t♦ ❞❡st❡s ❛ss✉♥t♦s t❛♥t♦ ❡♠ ❛✉❧❛s r❡❣✉❧❛r❡s q✉❛♥t♦ ❡♠ ❛✉❧❛s ❞❡ ❛♣r♦❢✉♥❞❛♠❡♥t♦ ♣❛r❛ ❛❧✉♥♦s ✐♥t❡r❡ss❛❞♦s✳

❖s ❞❛❞♦s ❤✐stór✐❝♦s ♥♦s q✉❛✐s ❢♦r❛♠ ❜❛s❡❛❞♦s ♦s t❡①t♦s ❛❝✐♠❛ ❢♦r❛♠ r❡t✐r❛❞♦s ❞♦ t❡①t♦ ✏▼❡♥❛❡❝♠✉s✱ ❆♣♦❧ô♥✐❝♦ ❡ ❛s ❙❡❝çõ❡s ❈ô♥✐❝❛s✑ ❞❡ ❍②❣✐♥♦ ❍✳ ❉♦♠✐♥❣✉❡s ✭■❊❩❩■✱ ●✳✱ ❋✉♥❞❛♠❡♥t♦s ❞❛ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ❱♦❧✳ ✼✱ ✷✵✵✺ ♣✳ ✺✽✲✺✾✮ ❡ ✏◗✉á❞r✐❝❛s✿ ❘❡s❡♥❤❛ ❍✐stór✐❝❛✑ ✭❱❊◆❚❯❘■◆■✱ ❏✳ ❏✳✱ ✷✵✵✸✱ ♣✳✶✻✶✲✶✻✸✮✳

✶✳✷ ●❡♦●❡❜r❛

❉❡♥tr❡ t❛♥t♦s s♦❢t✇❛r❡s ♣❛r❛ ❛✉①í❧✐♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞❛ ♠❛t❡♠át✐❝❛✱ ❡①♣❧✐❝♦ ♦ ♣♦rq✉❡ ❞❛ ❡s❝♦❧❤❛ ❞♦ ●❡♦●❡❜r❛ ♥❛ ❝♦♥❢❡❝çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

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●❡♦●❡❜r❛ ✶✼

q✉❡ r❡ú♥❡ ●❡♦♠❡tr✐❛✱ ➪❧❣❡❜r❛✱ P❧❛♥✐❧❤❛ ❞❡ ❈á❧❝✉❧♦✱ ●rá✜❝♦s✱ Pr♦❜❛❜✐❧✐❞❛❞❡✱ ❊st❛tís✲ t✐❝❛ ❡ ❈á❧❝✉❧♦s ❙✐♠❜ó❧✐❝♦s ❡♠ ✉♠ ú♥✐❝♦ ♣❛❝♦t❡ ❢á❝✐❧ ❞❡ s❡ ✉s❛r✳ ❖ ●❡♦●❡❜r❛ ♣♦ss✉✐ ✉♠❛ ❝♦♠✉♥✐❞❛❞❡ ❞❡ ♠✐❧❤õ❡s ❞❡ ✉s✉ár✐♦s ❡♠ ♣r❛t✐❝❛♠❡♥t❡ t♦❞♦s ♦s ♣❛ís❡s✳ ❖ ●❡♦●❡❜r❛ s❡ t♦r♥♦✉ ✉♠ ❧í❞❡r ♥❛ ár❡❛ ❞❡ s♦❢t✇❛r❡s ❞❡ ♠❛t❡♠át✐❝❛ ❞✐♥â♠✐❝❛✱ ❛♣♦✐❛♥❞♦ ♦ ❡♥s✐♥♦ ❡ ❛ ❛♣r❡♥❞✐③❛❣❡♠ ❡♠ ❈✐ê♥❝✐❛✱ ❚❡❝♥♦❧♦❣✐❛✱ ❊♥❣❡♥❤❛r✐❛ ❡ ▼❛t❡♠át✐❝❛✳✑ ❆❝✐♠❛ t❡♠♦s ❛ ❞❡s❝r✐çã♦ ♦✜❝✐❛❧ ❞♦ s♦❢t✇❛r❡ r❡t✐r❛❞❛ ♥♦ s✐t❡ ❞♦ ❞❡s❡♥✈♦❧✈❡❞♦r ❡♠ ❛❣♦st♦ ❞❡ ✷✵✶✻✳ ■♥❢❡❧✐③♠❡♥t❡ ♥ã♦ ✐r❡♠♦s ❡①♣❧♦r❛r t♦❞❛s ❛s ❢❡rr❛♠❡♥t❛s q✉❡ ♦ ♣r♦❣r❛♠❛ ❞❡s❡♥✈♦❧✈✐❞♦ ♣❡❧❛ ❡q✉✐♣❡ ❞❡ ▼❛r❦✉s ❍♦❤❡♥✇❛rt❡r ♥♦s ♦❢❡r❡❝❡✱ ♠❛s ✈❛♠♦s ❝♦♠♣r♦✈❛r s✉❛ ❢❛❝✐❧✐❞❛❞❡ ♥♦ ♠❛♥✉s❡❛♠❡♥t♦✱ s✉❛ ✈❡rs❛t✐❧✐❞❛❞❡ ❡ ♣♦❞❡r ❝♦♠♣✉t❛❝✐♦♥❛❧✳ ❖✉tr❛ ✈❛♥t❛❣❡♠ ❞♦ ●❡♦✲ ●❡❜r❛ é s❡✉ ❛❧❝❛♥❝❡✱ ♣♦r s❡r ✉♠ s♦❢t✇❛r❡ ♠✉❧t✐✲♣❧❛t❛❢♦r♠❛ ❡❧❡ ♣♦❞❡ s❡r ✐♥st❛❧❛❞♦ ❡♠ ♣r❛t✐❝❛♠❡♥t❡ t♦❞❛ ♠áq✉✐♥❛ ❝♦♠♣✉t❛❞♦r❛ ❡①✐st❡♥t❡ ❤♦❥❡✱ ♣♦✐s ♦ ♣r♦❣r❛♠❛ t❡♠ ✈❡rsõ❡s ♣❛r❛ ❛s ♣❧❛t❛❢♦r♠❛s ❉❡s❦t♦♣ ❲✐♥❞♦✇s✱ ▲✐♥✉① ❡ ▼❛❝ ❖s✳ ❊❧❡ t❛♠❜é♠ é ❡①❡❝✉tá✈❡❧ ❡♠ t❛❜❧❡ts ❝♦♠ s✐st❡♠❛ ♦♣❡r❛❝✐♦♥❛❧ ❆♥❞r♦✐❞✱ ❲✐♥❞♦✇s P❤♦♥❡ ❡ ✐❖❙✱ ❛❧é♠ ❞❡ ♦❢❡r❡❝❡r ✉♠❛ ❝❛❧❝✉❧❛❞♦r❛ ❣rá✜❝❛ ♠✉✐t♦ ✐♥t❡r❡ss❛♥t❡s ♣❛r❛ ❞✐s♣♦s✐t✐✈♦s ❆♥❞r♦✐❞✳ ❊♠ t♦❞❛s ❡ss❛s ♣❧❛t❛❢♦r♠❛s ♦ ♣r♦❣r❛♠❛ é ❞✐str✐❜✉í❞♦ ❞❡ ❢♦r♠❛ ❧✐✈r❡ ❡ ❣r❛t✉✐t❛✳

P♦rt❛♥t♦ ❛❝r❡❞✐t♦ q✉❡ ♦ ●❡♦●❡❜r❛ s❡rá ✉♠❛ ❣r❛♥❞❡ ❢❡rr❛♠❡♥t❛ ♥♦ ♣r♦❝❡ss♦ ❞❡ ♠♦❞❡r♥✐③❛çã♦ ❞♦ ❡♥s✐♥♦ q✉❡ ❡st❛♠♦s ♣❛ss❛♥❞♦ ❡ ❛❥✉❞❛♥❞♦ ❛ ❝♦♥str✉✐r✳

❆♦ ❜❛✐①❛r ♦ ●❡♦●❡❜r❛ ❡♠ ✇✇✇✳❣❡♦❣❡❜r❛✳♦r❣ ♣❛r❛ ❛❝♦♠♣❛♥❤❛r ❡st❡ t❡①t♦ ♥ã♦ ❞❡✐①❡ ❞❡ ❡①♣❧♦r❛r ♦s ♠❛t❡r✐❛✐s ❞✐s♣♦♥✐❜✐❧✐③❛❞♦s ❣r❛t✉✐t❛♠❡♥t❡ ♣❡❧♦s ✉s✉ár✐♦s ❞❛ ♣❧❛t❛✲ ❢♦r♠❛✳

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✶✽ ■♥tr♦❞✉çã♦

❋✐❣✉r❛ ✶✳✶✿ ❏❛♥❡❧❛ ♣❛❞rã♦ ❞♦ ●❡♦●❡❜r❛✳ ❉❡st❛❝❛❞♦s ♥❛ ✜❣✉r❛ ❛❝✐♠❛ t❡♠♦s ♦s s❡❣✉✐♥t❡s ✐t❡♥s✿

✶✳ ❇❛rr❛ ❞❡ ♠❡♥✉s ✷✳ ❇❛rr❛ ❞❡ ❢❡rr❛♠❡♥t❛s ✸✳ ❏❛♥❡❧❛ ❞❡ á❧❣❡❜r❛ ✹✳ ❏❛♥❡❧❛ ❞❡ ✈✐s✉❛❧✐③❛çã♦ ✺✳ ❈❛♠♣♦ ❞❡ ❡♥tr❛❞❛

✶✳✷✳✶ ❇❛rr❛ ❞❡ ♠❡♥✉s

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●❡♦●❡❜r❛ ✶✾

❋✉♥❝✐♦♥❛❧✐❞❛❞❡ ❈❛♠✐♥❤♦

❆❜r✐r ❛rq✉✐✈♦ ❆rq✉✐✈♦ ❆❜r✐r✳✳✳

❙❛❧✈❛r ❛rq✉✐✈♦ ❆rq✉✐✈♦ ●r❛✈❛r

❊①♣♦rt❛r ✜❣✉r❛ ❆rq✉✐✈♦ ❊①♣♦rt❛r ❏❛♥❡❧❛ ❞❡ ❱✐s✉❛❧✐③❛çã♦ ❝♦♠♦ ❬✳✳✳❪

❈❛s❛s ♣❛r❛ ❛rr❡❞♦♥❞❛♠❡♥t♦ ❖♣çõ❡s ❆rr❡❞♦♥❞❛♠❡♥t♦ n❝❛s❛s ❞❡❝✐♠❛✐s

▼✉❞❛r ✐❞✐♦♠❛ ❖♣çõ❡s ■❞✐♦♠❛

❆❥✉❞❛ ❆❥✉❞❛ ✏❚✐♣♦ ❞❡ ❛❥✉❞❛ ❞❡s❡❥❛❞❛✑

❖ ♠❡♥✉ ❊①✐❜✐r ❛t✐✈❛ ❡ ❞❡s❛t✐✈❛ ❢✉♥❝✐♦♥❛❧✐❞❛❞❡s ❡ ❝❛❧❝✉❧❛❞♦r❛s ❞♦ ♣r♦❣r❛♠❛✳

❋✐❣✉r❛ ✶✳✷✿ ▼❡♥✉ ❡①✐❜✐r✱ ❢✉♥❝✐♦♥❛❧✐❞❛❞❡s ❡ ❛t❛❧❤♦s✳

P♦r ♣❛❞rã♦✱ ♦ ●❡♦❣❡❜r❛ ❛♦ ❛❜r✐r ❡①✐❜❡ ❛♣❡♥❛s ❛ ❏❛♥❡❧❛ ❞❡ ➪❧❣❡❜r❛ ❏❛♥❡❧❛ ❞❡ ❱✐s✉❛❧✐③❛çã♦

❡ ♦ ❈❛♠♣♦ ❞❡ ❊♥tr❛❞❛✳ ◆♦ ❞❡❝♦rr❡r ❞♦ t❡①t♦ ✐r❡♠♦s ❝♦♥❤❡❝❡r ❡ tr❛❜❛❧❤❛r ❜❛st❛♥t❡ ❝♦♠ ❛ ❏❛♥❡❧❛ ❞❡ ❱✐s✉❛❧✐③❛çã♦ ✸❉ ❡ t❛♠❜é♠ ✐r❡♠♦s ❝♦♥❤❡❝❡r ❛ ❏❛♥❡❧❛ ❈❆❙✳ ❙✉❣❡r✐♠♦s q✉❡ ♦ ❧❡✐t♦r

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✷✵ ■♥tr♦❞✉çã♦

❋✐❣✉r❛ ✶✳✸✿ ❚❡❝❧❛❞♦s ✈✐rt✉❛✐s ❡♥❝♦♥tr❛❞♦s ❡♠ ❊①✐❜✐r ❚❡❝❧❛❞♦

✶✳✷✳✷ ❇❛rr❛ ❞❡ ❢❡rr❛♠❡♥t❛s

❆ ❜❛rr❛ ❞❡ ❢❡rr❛♠❡♥t❛s é ♦♥❞❡ s❡ ❡♥❝♦♥tr❛♠ ♦s ❜♦tõ❡s q✉❡ ♥♦s ❛❥✉❞❛♠ ❛ r❡❛❧✐③❛r ❝♦♥str✉çõ❡s ♦✉ ❝á❧❝✉❧♦s ❞❡ ❢♦r♠❛ ✐♥t✉✐t✐✈❛ ❡ ❣rá✜❝❛✳ ▼✉✐t♦s ❞♦s ❝♦♠❛♥❞♦s q✉❡ ✐r❡♠♦s ❛♣r❡s❡♥t❛r ♣♦❞❡♠ s❡r ❢❡✐t♦s ❛tr❛✈és ❞❡ ❜♦tõ❡s✱ ♠❛s ♦♣t❛♠♦s ♣♦r ✉s❛r ♦ ❝♦♠❛♥❞♦ ❞❡ t❡①t♦ ♣♦r s❡ ❡♥❝❛✐①❛r ♠❡❧❤♦r ❡♠ ♠í❞✐❛ ❡s❝r✐t❛✳

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●❡♦●❡❜r❛ ✷✶

✷ ❉✐♠❡♥sõ❡s

◆♦♠❡ í❝♦♥❡ ❈♦♠❛♥❞♦ ❞❡ t❡①t♦

P♦♥t♦ ❁◆❖▼❊P❖◆❚❖❃ ❂ ✭❁❝♦♦r❞ ①❃✱ ❁❝♦♦r❞ ②❃✮

P♦♥t♦ ❡♠ ❖❜❥❡t♦ P♦♥t♦❊♠❬ ❁❘❡❣✐ã♦❃ ❪

■♥t❡rs❡❝çã♦ ❞❡ ❉♦✐s ❖❜❥❡✲ t♦s

■♥t❡rs❡❝çã♦❬ ❁❖❜❥❡t♦❃✱ ❁❖❜❥❡t♦❃ ❪

P♦♥t♦ ▼é❞✐♦ ♦✉ ❈❡♥tr♦ P♦♥t♦▼é❞✐♦❬ ❁❙❡❣♠❡♥t♦❃ ❪

❘❡t❛ ❘❡t❛❬ ❁P♦♥t♦❃✱ ❁P♦♥t♦❃ ❪

❙❡❣♠❡♥t♦ ❙❡❣♠❡♥t♦❬ ❁P♦♥t♦❃✱ ❁P♦♥t♦❃ ❪

❱❡t♦r ❱❡t♦r❬ ❁P♦♥t♦ ■♥✐❝✐❛❧❃✱ ❁P♦♥t♦ ❋✐♥❛❧❃ ❪

❘❡t❛ P❡r♣❡♥❞✐❝✉❧❛r P❡r♣❡♥❞✐❝✉❧❛r❬ ❁P♦♥t♦❃✱ ❁❘❡t❛❃ ❪

❘❡t❛ P❛r❛❧❡❧❛ ❘❡t❛❬ ❁P♦♥t♦❃✱ ❁❘❡t❛ P❛r❛❧❡❧❛❃ ❪

▼❡❞✐❛tr✐③ ▼❡❞✐❛tr✐③❬ ❁❙❡❣♠❡♥t♦❃ ❪

❇✐ss❡tr✐③ ❇✐ss❡tr✐③❬ ❁P♦♥t♦❃✱ ❁P♦♥t♦❃✱ ❁P♦♥t♦❃ ❪

❘❡t❛ t❛♥❣❡♥t❡ ❚❛♥❣❡♥t❡❬ ❁P♦♥t♦ ❞❛ ❈✉r✈❛❃✱ ❁❈✉r✈❛❃ ❪ ❈ír❝✉❧♦ ❞❛❞♦s ❈❡♥tr♦ ❡

✉♠ ❞❡ s❡✉s P♦♥t♦s

❈ír❝✉❧♦❬ ❁P♦♥t♦❃✱ ❁P♦♥t♦❃ ❪

➶♥❣✉❧♦ ➶♥❣✉❧♦❬ ❁P♦♥t♦❃✱ ❁❱ért✐❝❡❃✱ ❁P♦♥t♦❃ ❪

◗✉❛♥❞♦ ❛ ❏❛♥❡❧❛ ❞❡ ❱✐s✉❛❧✐③❛çã♦ ✸❉ ❡st✐✈❡r ❛t✐✈❛✱ ❛❧❣✉♠❛s ❢❡rr❛♠❡♥t❛s ❡①❝❧✉s✐✈❛s ♣❛r❛ ❡ss❛ ❥❛♥❡❧❛ ✐rã♦ s❡r ❡①✐❜✐❞❛s✳ ❈♦♠♦ ❛s ❢❡rr❛♠❡♥t❛s sã♦ ♠✉✐t♦ ♣❛r❡❝✐❞❛s ✈♦✉ ❝♦❧♦❝❛r ❛♣❡♥❛s ❛❧❣✉♠❛s q✉❡ ✐r❡♠♦s ✉t✐❧✐③❛r✳

✸ ❉✐♠❡♥sõ❡s

◆♦♠❡ í❝♦♥❡ ❈♦♠❛♥❞♦ ❞❡ t❡①t♦

■♥t❡rs❡❝çã♦ ❞❡ ❉✉❛s s✉♣❡r❢í❝✐❡s ■♥t❡rs❡❝çã♦❬ ❁❖❜❥❡t♦❃✱ ❁❖❜❥❡t♦❃ ❪ P❧❛♥♦ ♣♦r três ♣♦♥t♦s P❧❛♥♦❬ ❁P♦♥t♦❃✱ ❁P♦♥t♦❃✱ ❁P♦♥t♦❃ ❪

P❧❛♥♦ P❡r♣❡♥❞✐❝✉❧❛r P❧❛♥♦P❡r♣❡♥❞✐❝✉❧❛r❬ ❁P♦♥t♦❃✱ ❁❘❡t❛❃ ❪ P❧❛♥♦ P❛r❛❧❡❧♦ ♦✉ ❈❡♥tr♦ P❧❛♥♦❬ ❁P♦♥t♦❃✱ ❁P❧❛♥♦❃ ❪

✶✳✷✳✸ ❏❛♥❡❧❛ ❞❡ á❧❣❡❜r❛

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✷✷ ■♥tr♦❞✉çã♦

❋✐❣✉r❛ ✶✳✹✿ ❏❛♥❡❧❛ ❞❡ ➪❧❣❡❜r❛ ❝♦♠ ❛❧❣✉♥s ❡❧❡♠❡♥t♦s✳

❘❡♣❛r❡ ♥❛ ✜❣✉r❛ ❛❝✐♠❛ q✉❡ ♦s ❡❧❡♠❡♥t♦s ✜❝❛♠ ♦r❣❛♥✐③❛❞♦s ♣♦r ❝❛t❡❣♦r✐❛✱ ♣♦❞❡♥❞♦ s❡r ♦❝✉❧t❛❞♦s ✭❝❧✐❝❛♥❞♦ ♥♦ s✐♥❛❧ ❞❡−✮ ♦✉ ❡①♣❛♥❞✐❞♦s ✭❝❧✐❝❛♥❞♦ ♥♦ s✐♥❛❧ ❞❡+✮✳ ❘❡♣❛r❡

t❛♠❜é♠ q✉❡ à ❡sq✉❡r❞❛ ❞❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❡①✐st❡ ✉♠❛ ❜♦❧✐♥❤❛ q✉❡ ♣♦❞❡ ❡st❛r ♣r❡❡♥✲ ❝❤✐❞❛ ♦✉ ♥ã♦✱ ❡❧❛ s✐♠❜♦❧✐③❛ q✉❡ é ♣♦ssí✈❡❧ ✈✐s✉❛❧✐③❛r ♦ ❡❧❡♠❡♥t♦ ❡♠ ❛❧❣✉♠❛ ❞❛s ❥❛♥❡❧❛s ❣rá✜❝❛s✱ s❡ ❛ ❜♦❧✐♥❤❛ ❡st✐✈❡r ✈❛③✐❛ s✐❣♥✐✜❝❛ q✉❡ ♦ ❡❧❡♠❡♥t♦ ❡stá ♦❝✉❧t♦✱ ❝❛s♦ ❝♦♥trár✐♦ ❡❧❡ ❡stá ✈✐sí✈❡❧✳

✶✳✷✳✹ ❏❛♥❡❧❛ ❞❡ ✈✐s✉❛❧✐③❛çã♦

❆ ❥❛♥❡❧❛ ❞❡ ✈✐s✉❛❧✐③❛çã♦ é ❛ ♣r✐♥❝✐♣❛❧ t❡❧❛ ❞♦ ●❡♦●❡❜r❛✱ ❡❧❛ s✐♠❜♦❧✐③❛ ✉♠ ♣❧❛♥♦ ✐♥✜♥✐t♦ ♦♥❞❡ ♣♦❞❡♠♦s ❝♦♥str✉✐r ♥♦ss♦s ❡❧❡♠❡♥t♦s✳ P❛r❛ ❛✉①✐❧✐❛r ♥❡ss❛s ❝♦♥str✉çõ❡s é ♣♦ssí✈❡❧ ❛t✐✈❛r ❡✐①♦s ❝❛rt❡s✐❛♥♦s ♦✉ ✉♠❛ ♠❛❧❤❛✱ q✉❡ ♣♦❞❡ s❡r ❝❛rt❡s✐❛♥❛✱ ✐s♦♠étr✐❝❛ ♦✉ ♣♦❧❛r✳

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●❡♦●❡❜r❛ ✷✸

❋✐❣✉r❛ ✶✳✺✿ ▼❡♥✉ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❏❛♥❡❧❛ ❞❡ ❱✐s✉❛❧✐③❛çã♦

✶✳✷✳✺ ❈❛♠♣♦ ❞❡ ❡♥tr❛❞❛

❖ ❝❛♠♣♦ ❞❡ ❡♥tr❛❞❛ é ❞❡st✐♥❛❞♦ ❛♦s ❝♦♠❛♥❞♦s ❞❡ t❡①t♦ ❞♦ s♦❢t✇❛r❡ ❝♦♠ ♦s q✉❛✐s ♣♦❞❡♠♦s ❡①❡❝✉t❛r ♦♣❡r❛çõ❡s ♦✉ ❝r✐❛r ♦❜❥❡t♦s✳

❖ ❝❛♠♣♦ ❛❝❡✐t❛ ❛ ❡♥tr❛❞❛ ❞❡ ✈ár✐♦s ❝❛r❛❝t❡r❡s ❡s♣❡❝✐❛✐s✱ ♣♦r ✐ss♦✱ q✉❛♥❞♦ ♦ ❝✉rs♦r ❡stá ❛t✐✈♦ ♥♦ ❝❛♠♣♦ ❛♣❛r❡❝❡ ♥♦ ✜♥❛❧ ❞❛ ❧✐♥❤❛ ✉♠ ❜♦tã♦ ❝♦♠ ♦ s✐♠❜♦❧♦ α ❡ q✉❛♥❞♦ ❝❧✐❝❛♠♦s ♥❡❧❡ sã♦ ❡①✐❜✐❞♦s ♦s ❝❛r❛❝t❡r❡s ❡s♣❡❝✐❛✐s ❛❝❡✐t♦s ♣❡❧♦ ♣r♦❣r❛♠❛✳

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✷ ❈ô♥✐❝❛s

◆♦ ❝❛♣ít✉❧♦ ❛ s❡❣✉✐r ❢❛r❡♠♦s ❛❧❣✉♠❛s ♦❜s❡r✈❛çõ❡s s♦❜r❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❡❧✐♣s❡✱ ♣❛rá❜♦❧❛ ❡ ❤✐♣ér❜♦❧❡✳ Pr✐♠❡✐r❛♠❡♥t❡ ❞❡✜♥✐r❡♠♦s ❝❛❞❛ ✉♠ ❞♦s ❡❧❡♠❡♥t♦s ❡ ❡♥tã♦ ♠♦s✲ tr❛r❡♠♦s ❝♦♠♦ ❣❡r❛r ♥♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛ ✉♠ ❝♦♥❡ ♣❛r❛ ✐♥t❡rs❡❝t❛r♠♦s ✉♠ ♣❧❛♥♦ ❞❡ ❢♦r♠❛ ❛ ❝r✐❛r ❝❛❞❛ ✉♠ ❞♦s ❡❧❡♠❡♥t♦s ❝✐t❛❞♦s ❡ ❡♠ s❡❣✉✐❞❛ ✈❛♠♦s ❝♦♥str✉✐r ❛s ✜❣✉r❛s ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ❛ ♣❛rt✐r ❞❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛✱ ✈❛r✐❛♥❞♦ ❡ ❡♥t❡♥❞❡♥❞♦ ♦s ♣❛râ♠❡tr♦s ❞❡ss❛s ❡q✉❛çõ❡s✳

✷✳✶ ❈✐r❝✉♥❢❡rê♥❝✐❛

❉❡✜♥✐çã♦ ✷✳✶✳ ❉❛❞♦s ✉♠ ♣♦♥t♦C = (a, b)✱ ♣❡rt❡♥❝❡♥t❡ ❛ ✉♠ ♣❧❛♥♦α❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ r > 0✱ ❝❤❛♠❛✲s❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❛♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s Pi ❞❡ α t❛✐s q✉❡ d(C, Pi) = r✱ ♦♥❞❡ ✉s✉❛❧♠❡♥t❡ ✉s❛♠♦s ❛ ❞✐stâ♥❝✐❛ ❡✉❝❧✐❞✐❛♥❛✳

✷✳✶✳✶ ❙❡çã♦ ❝ô♥✐❝❛

❆❧é♠ ❞❛ ❞❡✜♥✐çã♦ ❞❛❞❛ ❛❝✐♠❛✱ s❛❜❡♠♦s q✉❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ é ✉♠❛ s❡çã♦ ❝ô♥✐❝❛ ♦❜t✐❞❛ q✉❛♥❞♦ s❡❝❝✐♦♥❛♠♦s ✉♠ ❝♦♥❡ ❝♦♠ ✉♠ ♣❧❛♥♦ ♣❡r♣❡♥❞✐❝✉❧❛r à ❛❧t✉r❛ ❞❡st❡✳ P♦✲ ❞❡♠♦s ✉s❛r ♦ ●❡♦●❡❜r❛ ♣❛r❛ ♠♦str❛r ❛ ❝ô♥✐❝❛ s❡♥❞♦ ❣❡r❛❞❛✳

P❛r❛ ❣❡r❛r♠♦s ❛ s✐♠✉❧❛çã♦ ✈❛♠♦s ❝r✐❛r ♦s s❡❣✉✐♥t❡s ❡❧❡♠❡♥t♦s✿

• ❯♠ ♣❛râ♠❡tr♦ ✭♥ú♠❡r♦ r❡❛❧✮❀

• ❯♠ ♣❧❛♥♦ ❞❡♣❡♥❞❡♥t❡ ❞♦ ♣❛râ♠❡tr♦ ❝r✐❛❞♦❀ • ❯♠ ❝♦♥❡✳

❉❡♣♦✐s ✐r❡♠♦s ❛t✐✈❛r ♥♦ ●❡♦❣❡❜r❛ ✉♠❛ ❛♥✐♠❛çã♦ q✉❡ ♠♦str❛rá ♦ q✉❡ ❛❝♦♥t❡❝❡ ❛♦ ✈❛r✐❛r♠♦s ♦ ♣❛râ♠❡tr♦✳ ❈♦♠ ❛ ❥❛♥❡❧❛ ❞♦ ●❡♦●❡❜r❛ ❛❜❡rt❛ ✈❛♠♦s s❡❣✉✐r ♦ s❡❣✉✐♥t❡ ❝❛♠✐♥❤♦ ♥❛ ❜❛rr❛ ❞❡ ♠❡♥✉s ❊①✐❜✐r ❏❛♥❡❧❛ ✸❉ ❡ ❡♥tã♦ ❡①❡❝✉t❛r ♦s ♣❛ss♦s ❛❜❛✐①♦✳

P❛r❛ ❝r✐❛r ✉♠ ♣❛râ♠❡tr♦ r❡❛❧ ❞✐❣✐t❛♠♦s ♥♦ ❝❛♠♣♦ ❊♥tr❛❞❛ ♦ ❝♦♠❛♥❞♦✿

❛ ❂ ✶

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✷✻ ❈ô♥✐❝❛s

◆❛ ❏❛♥❡❧❛ ❞❡ ➪❧❣❡❜r❛ ✐rá ❛♣❛r❡❝❡r s✉❛ ❝r✐❛çã♦ ❝♦♠♦ ✈❡♠♦s ♥❛ ✜❣✉r❛ ❛❜❛✐①♦

❋✐❣✉r❛ ✷✳✶✿ P❛râ♠❡tr♦ ❝r✐❛❞♦✳ P♦❞❡♠♦s ♥♦♠❡á✲❧♦ ❝♦♠♦ ❞❡s❡❥❛r♠♦s✳ a✱ k✱ b✱cte✱ ❡t❝✳ P❛r❛ q✉❡ ❛♣❛r❡ç❛ ♦ ♣❧❛♥♦ ♣❛r❛❧❡❧♦ ❛ xy ✈❛♠♦s ❞✐❣✐t❛r ♥♦ ❝❛♠♣♦ ❞❡ ❊♥tr❛❞❛ ❛ s❡✲ ❣✉✐♥t❡ ❡q✉❛çã♦

③❂❛

❋✐❣✉r❛ ✷✳✷✿ ❊✐s ♦ ♣❧❛♥♦ ♥❛ ❛❧t✉r❛z ❡♠ q✉❡ a ❢♦✐ s❡t❛❞♦✱ ♥♦ ❝❛s♦ a= 1✳

❆❣♦r❛ ♣❛r❛ ✐♥❝❧✉✐r ♥❛ ❝♦♥str✉çã♦ ♦ ❝♦♥❡✱ ❡♥tr❡ ❝♦♠ ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦

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❈✐r❝✉♥❢❡rê♥❝✐❛ ✷✼

❋✐❣✉r❛ ✷✳✸✿ ❊s♣❛ç♦ ❝♦♠ ♦s ❡❧❡♠❡♥t♦s ❝r✐❛❞♦s✱ ✉♠ ❝♦♥❡ ❡ ✉♠ ♣❧❛♥♦ q✉❡ ♦ ✐♥t❡rs❡❝t❛✳

❈♦♠ ❛♠❜♦s ♦s ❡❧❡♠❡♥t♦s ❡♠ t❡❧❛✱ ♣♦❞❡♠♦s ❣✐r❛r ❛ ✜❣✉r❛ ❝❧✐❝❛♥❞♦ ❡♠ ✉♠ ❡s♣❛ç♦ ❡♠ ❜r❛♥❝♦ ❞❛ t❡❧❛ ❡✱ s❡♠ s♦❧t❛r ♦ ❜♦tã♦ ❞♦ ✧♠♦✉s❡✧✱ ♠♦✈ê✲❧♦ ❡♠ ✈ár✐❛s ❞✐r❡çõ❡s✳ ❇❛st❛ ❝❧✐❝❛r ♥♦ ❜♦tã♦ ●✐r❛r ❥❛♥❡❧❛ ❞❡ ✈✐s✉❛❧✐③❛çã♦ ✸❉ ♣❛r❛ ❡①❡❝✉t❛r ♦ ♠♦✈✐♠❡♥t♦✳

❋✐❣✉r❛ ✷✳✹✿ ❇♦tã♦ ♣❛r❛ ❣✐r♦ ❞♦ ❡s♣❛ç♦ ❡♠ t❡❧❛✳

▼✉✐t♦ ❜❡♠✱ ✈❛♠♦s ❢❛③❡r ♦ ♣❛râ♠❡tr♦ ✈❛r✐❛r ❞❡ ❢♦r♠❛ ❛✉t♦♠át✐❝❛ ❡ ✈✐s✉❛❧✐③❛r ♦ r❡s✉❧t❛❞♦✱ ♣❛r❛ ✐ss♦ ♥❛ ❏❛♥❡❧❛ ❞❡ ➪❧❣❡❜r❛ ❛♣♦♥t❡ ♦ ✧♠♦✉s❡✧❛♦ ♣❛râ♠❡tr♦✱ ❝❧✐q✉❡ ❝♦♠ ♦

❜♦tã♦ ❞✐r❡✐t♦ ❡ s❡❧❡❝✐♦♥❡ ❛ ♦♣çã♦ ❛♥✐♠❛r

❋✐❣✉r❛ ✷✳✺✿ ❆t✐✈❛♥❞♦ ✈❛r✐❛çã♦ ❛✉t♦♠át✐❝❛ ❞♦ ♣❛râ♠❡tr♦✳

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✷✽ ❈ô♥✐❝❛s

❋✐❣✉r❛ ✷✳✻✿ ❇♦tã♦ ♣❛r❛ ❣❡r❛r ✐♥t❡rs❡❝çã♦ ❡♥tr❡ s✉♣❡r❢í❝✐❡s✳

Pr✐♠❡✐r❛♠❡♥t❡ ❝❧✐❝❛♠♦s ❡♠ ■♥t❡rs❡❝çã♦ ❞❡ ❞✉❛s s✉♣❡r❢í❝✐❡s ❡♠ s❡❣✉✐❞❛ ❝❧✐❝❛♠♦s ♥♦ ❝♦♥❡

❡ ❞❡♣♦✐s ♥♦ ♣❧❛♥♦✳ ✭❚❛♠❜é♠ ♣♦❞❡♠♦s ❢❛③❡r ❛ s❡❧❡çã♦ ❞❡st❡s ♦❜❥❡t♦s ❛tr❛✈és ❞❛ ❏❛♥❡❧❛ ❞❡ á❧❣❡❜r❛✮✳

➱ ♣♦ssí✈❡❧ ♦❜s❡r✈❛r q✉❡ ❛♣❛r❡❝❡✉ t❛♥t♦ ♥❛ ❏❛♥❡❧❛ ✸❉ q✉❛♥t♦ ♥❛ ❏❛♥❡❧❛ ❞❡ ➪❧❣❡❜r❛ ♦ ♥♦✈♦

❡❧❡♠❡♥t♦ ❥á ❝❧❛ss✐✜❝❛❞♦ ❝♦♠♦ ❝ô♥✐❝❛ ❡ t❛♠❜é♠ s✉❛ ❡q✉❛çã♦✳

❋✐❣✉r❛ ✷✳✼✿ ■♥t❡rs❡❝çã♦ ❡♥tr❡ ♦ ❝♦♥❡ ❡ ♦ ♣❧❛♥♦✳

P❛r❛ ✜❝❛r ♠❛✐s ❝❧❛r♦ ♣♦❞❡♠♦s ♦❝✉❧t❛r ♦ ❝♦♥❡ ❡✴♦✉ ♦ ♣❧❛♥♦ ❞❡✐①❛♥❞♦ ❡①♣♦st♦ ❛♣❡♥❛s ♦ r❡s✉❧t❛❞♦ ❞❛ ✐♥t❡rs❡❝çã♦✳

❋✐❣✉r❛ ✷✳✽✿ P❛r❛ ♦❝✉❧t❛r ♦ ♦❜❥❡t♦ ❜❛st❛ ❞❡s♠❛r❝❛r ❛ ❧✐♥❤❛ ❊①✐❜✐r ♦❜❥❡t♦

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❈✐r❝✉♥❢❡rê♥❝✐❛ ✷✾

❋✐❣✉r❛ ✷✳✾✿ ❊①✐❜✐♥❞♦ ❛♣❡♥❛s ♦ r❡s✉❧t❛❞♦ ❞❛ ✐♥t❡rs❡❝çã♦✳

✷✳✶✳✷ ❊q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛

❈♦♠♦ é ✉♠❛ ❞❡♠♦♥str❛çã♦ ❜❡♠ s✐♠♣❧❡s✱ ✈❛♠♦s ❞❡❞✉③✐r ❛q✉✐ q✉❡ ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❛ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ C = (a, b)❡ r❛✐♦ r q✉❡ t❡♠ ♣♦r s✉❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛

(xa)2+ (yb)2 =r2 ✭✷✳✶✮ ❆ ❡q✉❛çã♦ ♥♦r♠❛❧❧ é ❞❛❞❛ ♣♦r

x2+y2+2ax+2bx+a2+b2r2 = 0 ✭✷✳✷✮

❉❡♠♦♥str❛çã♦ ❙❡❥❛ P ✉♠ ♣♦♥t♦ (x, y) ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ P❡❧❛ ❞❡✜♥✐çã♦✱ t❡♠♦s

q✉❡ ❛ ❞✐stâ♥❝✐❛ P C é ✐❣✉❛❧ ❛♦ r❛✐♦ r✳ ❆♣❧✐❝❛♥❞♦ ❛ ❢ór♠✉❧❛ ♣❛r❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡st❡s ♣♦♥t♦s t❡♠♦s

p

(xa)2+ (yb)2 =r(xa)2+ (y

−b)2 =r2 ✭✷✳✸✮ ❈♦♥❝❧✉✐♥❞♦ ❛ss✐♠ ❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ C = (a, b) ❡ r❛✐♦

r✳ P❛r❛ ♦❜t❡r♠♦s ❛ ❡q✉❛çã♦ ♥♦r♠❛❧✱ ❜❛st❛ ❡①♣❛♥❞✐r ❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛✳

P❛r❛ ✐♥t✉✐r ❞❡ ❢♦r♠❛ ❝❧❛r❛ ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ❝❛❞❛ ✉♠ ❞❡st❡ ❡❧❡♠❡♥t♦s✱ ❛ ❛❥✉❞❛ ❞❡ ✉♠ s♦❢t✇❛r❡ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛ ❝♦♠♦ ♦ ●❡♦●❡❜r❛ ❞❡✐①❛ ❛ t❛r❡❢❛ ❜❡♠ s✐♠♣❧❡s✳ ❆ss✐♠ é ♣♦ssí✈❡❧ q✉❡ ♦s ❛❧✉♥♦s ❞❡❞✉③❛♠ ❞❡ ❢♦r♠❛ ❡♠♣ír✐❝❛ ❝♦♠♦ ♦s ❡❧❡♠❡♥t♦s a✱b ❡ r ✐♥✢✉❡♥❝✐❛♠ ♥❛ ✜❣✉r❛ ❣❡♦♠étr✐❝❛ ❣❡r❛❞❛ ♣❡❧❛ ❡q✉❛çã♦✳

P❛r❛ r❡❛❧✐③❛r ❛ ❝♦♥str✉çã♦✱ ❡♥tr❡ ♥♦ ●❡♦●❡❜r❛✱ ❞❡st❛ ✈❡③ s❡♠ ❛ ❏❛♥❡❧❛ ✸❉ ❛t✐✈❛✱

♣❛r❛ ❝r✐❛r ♦s ♣❛râ♠❡tr♦s ✐♥t❡r❛t✐✈♦s✳ ◆♦ ❝❛♠♣♦ ❞❡ ❊♥tr❛❞❛ ❞♦ ♣r♦❣r❛♠❛ ✈❛♠♦s ❞✐❣✐t❛r

♦s s❡❣✉✐♥t❡s ❝♦♠❛♥❞♦s✿

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✸✵ ❈ô♥✐❝❛s

■rá ❛♣❛r❡❝❡r ❡♠ s✉❛ ❏❛♥❡❧❛ ❞❡ ➪❧❣❡❜r❛ ❛ ❝❛t❡❣♦r✐❛ ✏◆ú♠❡r♦✑ ❝♦♠ ♦s três ♣❛râ♠❡tr♦

❝r✐❛❞♦s✳ ❘❡♣❛r❡ q✉❡ ♦ ♣❡q✉❡♥♦ ❝✐r❝✉❧♦ à ❡sq✉❡r❞❛ ❞❡ ❝❛❞❛ ♦❜❥❡t♦ ♣r♦✈❛✈❡❧♠❡♥t❡ ❡st❛rá ✈❛③✐♦✱ ❝♦♠♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✳

❋✐❣✉r❛ ✷✳✶✵✿ ❖❜❥❡t♦s ♦❝✉❧t♦s ♥❛ ❏❛♥❡❧❛ ❞❡ ❱✐s✉❛❧✐③❛çã♦✳

❈❧✐❝❛♥❞♦ ♥❡st❡ ♣❡q✉❡♥♦ ❝ír❝✉❧♦ ♦✉ ♥♦ ♦❜❥❡t♦ ❝♦♠ ♦ ❜♦tã♦ ❞✐r❡✐t♦ ❡ ❞❡♣♦✐s ❡♠

❊①✐❜✐r ♦❜❥❡t♦ ✐rá ❛♣❛r❡❝❡r ✉♠ ❝♦♥tr♦❧❡ ❞❡s❧✐③❛♥t❡ ❝♦♠ ❝❛❞❛ ✉♠ ❞♦s ♣❛râ♠❡tr♦s✳

❋✐❣✉r❛ ✷✳✶✶✿ ❈♦♥tr♦❧❡s ❞❡s❧✐③❛♥t❡s ♣❛r❛ ❛❧t❡r❛çã♦ ❞❡ ✈❛❧♦r❡s✳

❆♦ ❞✐❣✐t❛r♠♦s ❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♥♦ ❝❛♠♣♦ ❞❡ ❊♥tr❛❞❛

✭①✲❛✮❫✷✰✭②✲❜✮❫✷❂r❫✷

✐rá ❛♣❛r❡❝❡r ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ C = (0,0)❡ r❛✐♦ r= 1 ❡ ♥❛ ❏❛♥❡❧❛ ❞❡ ➪❧❣❡❜r❛

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❈✐r❝✉♥❢❡rê♥❝✐❛ ✸✶

❋✐❣✉r❛ ✷✳✶✷✿ ❊q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ s❡✉ r❡s✉❧t❛❞♦✳

❉❡s❧✐③❡ ♦ ❝♦♥tr♦❧❡ ♣❛r❛ ❛❧t❡r❛r ♦s ♣❛râ♠❡tr♦s ❡ s❡rá ❢á❝✐❧ ♥♦t❛r ❝♦♠♦ ❝❛❞❛ ✉♠ ❞❡❧❡s ✐rá ✐♥✢✉❡♥❝✐❛r ♥❛ ✜❣✉r❛✱ ♦ a ❞❡✈❡rá ❞❡s❧♦❝❛r ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❤♦r✐③♦♥t❛❧♠❡♥t❡✱ ♦ b ✈❡rt✐❝❛❧♠❡♥t❡ ❡ ♦ r ❛✉♠❡♥t❛r ♦✉ ❞✐♠✐♥✉✐r ♦ r❛✐♦✳ P♦❞❡ ❢❛❝✐❧✐t❛r ❛ ✈✐s✉❛❧✐③❛çã♦ ❞♦ ❡s♣❡❝t❛❞♦r s❡ ❛ ♠❛❧❤❛ ❡st✐✈❡r ❡①♣♦st❛ ♥♦ ♣❧❛♥♦ ❡ s❡ ❝r✐❛r♠♦s ✉♠ ♣♦♥t♦ ❝♦♠ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ❝❡♥tr♦✳

P❛r❛ ❝r✐❛r ✉♠ ♣♦♥t♦ ❝♦♠ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ❝❡♥tr♦ ✈❛♠♦s ❞✐❣✐t❛r ♥♦ ❝❛♠♣♦ ❞❡

❊♥tr❛❞❛

❈❂✭❛✱❜✮

❖ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❛♣❛r❡❝❡rá ❡ ✐rá ❛❝♦♠♣❛♥❤á✲❧❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s ♠✉❞❛♥✲ ç❛s✱ ♣♦✐s ♦ ❞❡✐①❛♠♦s ❞❡♣❡♥❞❡♥t❡ ❞♦s ♠❡s♠♦s ♣❛râ♠❡tr♦s✳

❙❡ q✉✐s❡r♠♦s ❞❡✐①❛r ❛✐♥❞❛ ♠❛✐s ❝❧❛r♦✱ ♣♦❞❡♠♦s ♣❡❞✐r ♣❛r❛ q✉❡ ♦ rót✉❧♦ ❞❡ss❡ ♣♦♥t♦ s❡❥❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❞♦ ♠❡s♠♦✳ P❛r❛ ✐ss♦ ❜❛st❛ ❝❧✐❝❛r ♥♦ ♣♦♥t♦ C ❝♦♠ ♦ ❜♦tã♦ ❞✐r❡✐t♦ ❡ ❡♥tã♦ Pr♦♣r✐❡❞❛❞❡s ❊①✐❜✐r rót✉❧♦ ❞❡✐①❡ ♠❛r❝❛❞♦ ♦ q✉❛❞r❛❞✐♥❤♦ à s✉❛

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✸✷ ❈ô♥✐❝❛s

❋✐❣✉r❛ ✷✳✶✸✿ ❏❛♥❡❧❛ ♦♥❞❡ ❝♦♥✜❣✉r❛♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♦❜❥❡t♦s✳

❋✐❣✉r❛ ✷✳✶✹✿ ❈♦♦r❞❡♥❛❞❛s s❡♥❞♦ ❡①✐❜✐❞❛s ❝♦♠♦ rót✉❧♦ ❞♦ ♣♦♥t♦✳

P♦❞❡♠♦s ❛❞♦t❛r ♦ ♠❡s♠♦ ♣r♦❝❡❞✐♠❡♥t♦ ❝♦♠ ❛ ❡q✉❛çã♦ ❣❡r❛❧✱ ♠❛s ♥❛ ❏❛♥❡❧❛ ❞❡ ➪❧❣❡❜r❛

é ♣r♦✈á✈❡❧ q✉❡ ♦ ♣r♦❣r❛♠❛ ❡①✐❜❛ ❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛✳ P❛r❛ ❛❧t❡r❛r ♦ ♠♦❞♦ ❞❡ ❡①✐❜✐çã♦ ❜❛st❛ ❝❧✐❝❛r ♥❛ ❡q✉❛çã♦ ❝♦♠ ♦ ❜♦tã♦ ❞✐r❡✐t♦ ❡ ❞❡♣♦✐s ♥❛ ♦♣çã♦ ✏❊q✉❛çã♦ ax2 +bxy+

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❊❧✐♣s❡ ✸✸

❋✐❣✉r❛ ✷✳✶✺✿ ▼✉❞❛♥❞♦ ♦ ❢♦r♠❛t♦ ❞❡ ❡①✐❜✐çã♦ ❞❛ ❡q✉❛çã♦✳

✷✳✷ ❊❧✐♣s❡

❉❡✜♥✐çã♦ ✷✳✷✳ ❉❛❞♦s ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s F1 ❡ F2 ❝❤❛♠❛❞♦s ❞❡ ❢♦❝♦s✱ ❛ ❡❧✐♣s❡ é ♦

❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s P ❞♦ ♣❧❛♥♦ ❝✉❥❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❛ F1 ❡ F2 ✱ d(F1, F2)✱ é ✐❣✉❛❧

❛ ✉♠❛ ❝♦♥st❛♥t❡ 2a >0✳ ❖✉ s❡❥❛✱ s❡ d(F1, F2) = 2c✱ t❡♠♦s q✉❡

elipse={P |d(P, F1) +d(P, F2) = 2a} ✭✷✳✹✮

✷✳✷✳✶ ❙❡çã♦ ❝ô♥✐❝❛

P❛r❛ ❝r✐❛r ❛ ✐♥t❡rs❡❝çã♦ ❞❡ ✉♠ ♣❧❛♥♦ ❝♦♠ ✉♠ ❝♦♥❡ ❝❡♥tr❛❞♦ ♥❛ ✐♥t❡rs❡❝çã♦ ❞♦s ❡✐①♦s xyz ❣❡r❛♥❞♦ ✉♠❛ ❡❧✐♣s❡ ✈❛♠♦s ❝r✐❛r ♦s s❡❣✉✐♥t❡s ❡❧❡♠❡♥t♦s✿

• ❯♠ ♣❛râ♠❡tr♦ ✭♥ú♠❡r♦ r❡❛❧✮❀

• ❯♠ ✈❡t♦r ♥♦r♠❛❧ q✉❡ ❞❡✜♥❡ ♦ ♣❧❛♥♦❀ • ❯♠ ♣♦♥t♦ P ❞❡♣❡♥❞❡♥t❡ ❞♦ ♣❛râ♠❡tr♦❀

• ❖ ♣❧❛♥♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ✈❡t♦r ♥♦r♠❛❧ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦P❀

• ❯♠ ❝♦♥❡✳

❉❡♣♦✐s ❞✐ss♦ ✈❛♠♦s ❛t✐✈❛r ♥♦ ●❡♦❣❡❜r❛ ✉♠❛ ❛♥✐♠❛çã♦ q✉❡ ♠♦str❛ ♦ q✉❡ ❛❝♦♥t❡❝❡ ❛♦ ✈❛r✐❛r♠♦s ♦ ♣❛râ♠❡tr♦✳

❉❡st❛ ✈❡③ ♥ã♦ ✈❛♠♦s ❞❡t❛❧❤❛r t❛♥t♦ ♥❛s ✐♠❛❣❡♥s✱ ♠❛s ♠❛♥t❡r❡♠♦s ♦ ♣❛ss♦ ❛ ♣❛ss♦ ❝♦♠ ♦s ❝♦♠❛♥❞♦s✳

❉❡✐①❡ ♦ ●❡♦●❡❜r❛ ❝♦♠ ❛ ❏❛♥❡❧❛ ✸❉ ❛❜❡rt❛ ❡ ✈❛♠♦s ❣❡r❛r ♦s ♦❜❥❡t♦s✳

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✸✹ ❈ô♥✐❝❛s

❝❂✶

P❛r❛ ❝r✐❛r ♦ ✈❡t♦r ♥♦r♠❛❧ ✐♥s❡r✐♠♦s ♦ ❝♦♠❛♥❞♦

◆❂✭✶✱✵✱✷✮

P❛r❛ ❝r✐❛r ♦ ♣♦♥t♦ P ❝♦♠ ❝♦♦r❞❡♥❛❞❛ z ❞❡♣❡♥❞❡♥t❡ ❞♦ ♣❛râ♠❡tr♦ ✈❛♠♦s ❞✐❣✐t❛r

P❂✭✵✱✵✱❝✮

P❛r❛ ❝r✐❛r♠♦s ✉♠ ♣❧❛♥♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ✈❡t♦rN q✉❡ ♣❛ss❛ ♣♦rP✱ t❡♠♦s ❛❧❣✉♠❛s ♦♣çõ❡s ❞❡ ❛❜♦r❞❛❣❡♠✳

▲❡♠❜r❛♠♦s q✉❡ N é✱ ♣♦r ❝♦♥str✉çã♦✱ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ q✉❡ ✐r❡♠♦s ❣❡r❛r✱ ♣♦rt❛♥t♦ t❛♠❜é♠ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ q✉❛❧q✉❡r ✈❡t♦r ♣❡rt❡♥❝❡♥t❡ ❛ ❡st❡ ♣❧❛♥♦✱ ❛ss✐♠ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♥tr❡ N ❡ −−→P P0 ✭s❡♥❞♦P0 ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞♦ ♣❧❛♥♦✮ é ♥✉❧♦✳

P♦rt❛♥t♦ s❡ N = (a, b, c)✱ P = (x, y, z) ❡ P0 = (x0, y0, z0)✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ♦

♣❧❛♥♦ ❡♠ q✉❡stã♦ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ♣❡❧❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦

a(xx0) +b(y−y0) +c(z−z0) = 0 ✭✷✳✺✮

❊ ✉♠❛ ♠❛♥❡✐r❛ s✐♠♣❧❡s ❞❡ ❞❡✜♥✐r♠♦s ✐ss♦ ♣❛r❛ ♦ ●❡♦●❡❜r❛ é ❞✐❣✐t❛♥❞♦ ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦ ♥♦ ❝❛♠♣♦ ❞❡ ❊♥tr❛❞❛✿

①✭◆✮✯✭① ✲ ①✭P✮✮ ✰ ②✭◆✮✯✭② ✲ ②✭P✮✮ ✰ ③✭◆✮✯✭③ ✲ ③✭P✮✮ ❂ ✵

❘❡♣❛r❡ ♥♦s ❝♦♠❛♥❞♦s ①✭◆✮✱ ②✭◆✮✱ ③✭◆✮✱ ①✭P✮✱ ②✭P✮✱ ③✭P✮ q✉❡ s❡r✈❡♠ ♣❛r❛ ❝❤❛♠❛r ❛s ❝♦♦r❞❡♥❛❞❛s x✱y ❡ z ❞❡N ❡ P✳

❯♠ ♦✉tr♦ ♠♦❞♦ ❞❡ ❝♦♥str✉✐r ♦ ♣❧❛♥♦ é ❝♦♠ ♦s ❝♦♠❛♥❞♦s✿

✈ ❂ ✈❡t♦r❬◆❪

P❧❛♥♦P❡r♣❡♥❞✐❝✉❧❛r❬P✱ ✈❪

♦ ❝♦♠❛♥❞♦ ✈ ❂ ✈❡t♦r❬◆❪ ❛tr❡❧❛ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ N ❛ ✉♠❛ ❡♥t✐❞❛❞❡ ✈❡t♦r✐❛❧✱ ❛♥t❡s ❞✐ss♦N ❡r❛ tr❛t❛❞♦ ❝♦♠♦ ✉♠ ♣♦♥t♦✱ t❛♥t♦ q✉❡ ❞❡♣♦✐s ❞❡ ❢❛③ê✲❧♦ ❛♣❛r❡❝❡r ✉♠❛ ✢❡✲ ❝❤❛ ♣❛r❛ r❡♣r❡s❡♥t❛rv ❡N ❝♦♥t✐♥✉❛ r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ ♣❡q✉❡♥❛ ❡s❢❡r❛✳ P♦❞❡rí❛♠♦s ❞❡s❞❡ ♦ ✐♥í❝✐♦ t❡r ❞❡❝❧❛r❛❞♦N ❝♦♠♦ ✉♠ ✈❡t♦r ❛♦ ✐♥✈és ❞❡ ✉♠ ♣♦♥t♦ s❡ ♦ ♥♦♠❡❛ss❡ ❝♦♠ ✉♠❛ ❧❡tr❛ ♠✐♥ús❝✉❧❛ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♥ ❂ ✭✶✱✵✱✶✮✱ ♣♦ré♠ ♣❡r❞❡r✐❛ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ♠❛♥✐♣✉❧❛çã♦ ❞♦ ✐t❡♠ ♥❛ ❏❛♥❡❧❛ ✸❉✳

❖ ❝♦♠❛♥❞♦ P❧❛♥♦P❡r♣❡♥❞✐❝✉❧❛r❬❁P♦♥t♦❃✱ ❁❱❡t♦r❃❪ ❣❡r❛ ♦ ♣❧❛♥♦ ♣❡r♣❡♥❞✐✲ ❝✉❧❛r ❛♦ ✈❡t♦r✱ ♥♦ ❝❛s♦ v✱ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ ❡♠ q✉❡stã♦✱ ♥♦ ❝❛s♦P✳

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❊❧✐♣s❡ ✸✺

❉❛q✉✐ ♣❛r❛ ❢r❡♥t❡✱ ♣❛r❛ ❣❡r❛r ♦ ♣❧❛♥♦ ♥♦ ●❡♦●❡❜r❛✱ ✐r❡♠♦s ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ q✉❡ ♦ ❞❡s❝r❡✈❡✱ ♠❛s ✜❝❛ ❛ ❝r✐tér✐♦ ❞♦ ❧❡✐t♦r ❡①❡❝✉t❛r ❛ ❛çã♦ ❞❡ ♦✉tr❛ ♠❛♥❡✐r❛ q✉❡ ❣❡r❡ ✉♠ r❡s✉❧t❛❞♦ ❡q✉✐✈❛❧❡♥t❡✳

❈♦♥t✐♥✉❛♥❞♦ ❛ ❝♦♥str✉çã♦✱ ✈❛♠♦s ❝r✐❛r ♦ ❝♦♥❡ ❝❡♥tr❛❞♦ ♥❛ ✐♥t❡rs❡❝çã♦ ❞❡xyz ❞✐❣✐✲ t❛♥❞♦✿

①❫✷ ✰ ②❫✷ ❂ ③❫✷

❆♣ós ♦ ♣r♦❝❡❞✐♠❡♥t♦ t❡r❡♠♦s ❝♦♠♦ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐♥t❡ ❝♦♥✜❣✉r❛çã♦✿

❋✐❣✉r❛ ✷✳✶✻✿ ■♥t❡rs❡❝çã♦ ❡♥tr❡ ♦ ❝♦♥❡ ❡ ♦ ♣❧❛♥♦ ❣❡r❛♥❞♦ ✉♠❛ ❡❧✐♣s❡✳

P♦❞❡♠♦s r♦t❛❝✐♦♥❛r ❛ ✜❣✉r❛ ❝♦♠ ❛ ❛❥✉❞❛ ❞♦ ❜♦tã♦ ❞❡ ❣✐r♦ ❡ é ♣♦ssí✈❡❧ ♥♦t❛r ❛ s❡çã♦ ❝ô♥✐❝❛ ❡♠ ❛③✉❧✳ P❛r❛ ❞❛r ✉♠ ♠❛✐♦r ❞❡st❛q✉❡ à ❡❧✐♣s❡ ✈❛♠♦s ✉s❛r ♦ ❜♦tã♦

■♥t❡rs❡❝çã♦ ❡♥tr❡ ❞✉❛s s✉♣❡r❢í❝✐❡s ❡ ❝❧✐❝❛r ♥♦ ❝♦♥❡ ❡ ♥♦ ♣❧❛♥♦✳ ❈♦♠♦ r❡s✉❧t❛❞♦ t❡r❡♠♦s ❡♠

❞❡st❛q✉❡ ❛ ❡❧✐♣s❡ ❡♠ q✉❡stã♦✳

❈❧✐❝❛♥❞♦ ❝♦♠ ♦ ❜♦tã♦ ❞✐r❡✐t♦ ♥♦ ♣❛râ♠❡tr♦c❡ ❡♠ s❡❣✉✐❞❛ ❡♠ ❆♥✐♠❛r é ♣♦ssí✈❡❧ ✈❡r

♦s ✈ár✐♦s ❝♦rt❡s ♣❛r❛❧❡❧♦s ♥♦ ❝♦♥❡✳

❯♠❛ s✉❣❡stã♦ ❞❡ ✈✐s✉❛❧✐③❛çã♦✿ ❝❧✐q✉❡ ❝♦♠ ♦ ❜♦tã♦ ❞✐r❡✐t♦ ♥❛ ❏❛♥❡❧❛ ❞❡ ➪❧❣❡❜r❛ ♥❛

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✸✻ ❈ô♥✐❝❛s

❋✐❣✉r❛ ✷✳✶✼✿ ❈r✐❛♥❞♦ ✈✐st❛ ✷❉ ❞❛ ❡❧✐♣s❡ ❣❡r❛❞❛ ♣❡❧❛ ✐♥t❡rs❡❝çã♦✳

❖ ●❡♦●❡❜r❛ ✐rá ❛❞✐❝✐♦♥❛r ✉♠❛ ♥♦✈❛ ❥❛♥❡❧❛ ❝♦♠ ❛ ✈✐st❛ ❞♦ ♣❧❛♥♦ s❡❧❡❝✐♦♥❛❞♦✳ ❙❡ ♣❡❞✐r♠♦s ♣❛r❛ ❛♥✐♠❛r ♥♦✈❛♠❡♥t❡ ♦ ♣❛râ♠❡tr♦ c✱ s❡rá ♣♦ssí✈❡❧ ✈❡r ❛s ♠✉❞❛♥ç❛s q✉❡ ❛ ❡❧✐♣s❡ s♦❢r❡ ❝♦♠ ❛ ✈❛r✐❛çã♦✳

✷✳✷✳✷ ❊q✉❛çã♦ ❞❛ ❡❧✐♣s❡

❆ ❡q✉❛çã♦ r❡❞✉③✐❞❛ ❞❛ ❡❧✐♣s❡ é ❞❛❞❛ ♣♦r

(xm)2

a2 +

(yn)2

b2 = 1, ✭✷✳✻✮

❝♦♠ a ❡ b ♣♦s✐t✐✈♦s ❡ (m, n) ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ❝❡♥tr♦ ❞❛ ❡❧✐♣s❡✳ ❆ss✐♠ ❝♦♠♦ ♥❛

❝✐r❝✉♥❢❡rê♥❝✐❛ ♣♦❞❡♠♦s ❣❡r❛r ❝♦♥tr♦❧❡s ❞❡s❧✐③❛♥t❡s ♣❛r❛ a✱ b✱ m ❡ n ❡ ✈❡r ❝♦♠♦ ❡st❡s ❛❢❡t❛♠ ❛ ❡❧✐♣s❡✳ P❛r❛ ✐ss♦ ✈❛♠♦s ❞✐❣✐t❛r ♥♦ ●❡♦●❡❜r❛ ❛ s❡❣✉✐♥t❡ r♦t✐♥❛✿

◆♦ ❝❛♠♣♦ ❞❡ ❊♥tr❛❞❛ ✈❛♠♦s ❞✐❣✐t❛r

❛❂✷ ❜❂✶ ♠❂✶ ♥❂✶

✭①✲♠✮❫✷✴❛❫✷ ✰ ✭②✲♥✮❫✷✴❜❫✷ ❂ ✶

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❊❧✐♣s❡ ✸✼

❋✐❣✉r❛ ✷✳✶✽✿ ❊❧✐♣s❡ (x−m)2

a2 +

(y−n)2

b2 = 1 ❝♦♠ a= 2 ❡ b=m=n = 1✳

❆❧t❡r❛♥❞♦ ♦ ✈❛❧♦r ❞❡ m é ♣♦ssí✈❡❧ ✈❡r ✉♠❛ tr❛♥s❧❛çã♦ ♥❛ ❞✐r❡çã♦ ❞♦ ❡✐①♦ x❡ ❞❡s❧✐✲ ③❛♥❞♦ ♦ ❝♦♥tr♦❧❡ ❞♦ ♣❛râ♠❡tr♦ né ♣♦ssí✈❡❧ ❞❡s❧♦❝❛r ❛ ❡❧✐♣s❡ ♥❛ ❞✐r❡çã♦ ❞❛s ♦r❞❡♥❛❞❛s✳ ❊ ❛♦ ❛❧t❡r❛r♠♦s a ❡b ♠♦❞✐✜❝❛♠♦s ♦ t❛♠❛♥❤♦ ❞♦s ❡✐①♦s ❞❛ ❡❧✐♣s❡✳ ❉❛ ❢♦r♠❛ q✉❡ ❛ ❡q✉❛çã♦ ❡stá ♠♦♥t❛❞❛✱ a ❛❧t❡r❛ ♦ ❡✐①♦ ♣❛r❛❧❡❧♦ ❛♦ ❡✐①♦ ❝❛rt❡s✐❛♥♦ x ❡ b✱ ♦y✳

P❛r❛ ♠❛r❝❛r ♦s ❢♦❝♦s ❞❛ ❡❧✐♣s❡ ❧❡♠❜r❛♠♦s ❞❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡a2 =b2+c2 ❡♠

q✉❡ c é ♠❡t❛❞❡ ❞❛ ❞✐stâ♥❝✐❛ ❢♦❝❛❧✳ ❆ss✐♠ c = √a2b2✱ ♣♦ré♠ s❡ ♦ ❡✐①♦ ✈❡rt✐❝❛❧ ❢♦r

♠❛✐♦r q✉❡ ♦ ❤♦r✐③♦♥t❛❧ t❡r❡♠♦s q✉❡ c=√b2a2✳

❚❡♥❞♦ ✐ss♦ ❡♠ ♠❡♥t❡✱ ✈❛♠♦s ❞❡✜♥✐r ♦s ❢♦❝♦s ❞❛ ❡❧✐♣s❡ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✳ ❙❡ ♦ ❡✐①♦ ♠❛✐♦r ❢♦r ♦ ❤♦r✐③♦♥t❛❧ ❡♥tã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦s ❢♦❝♦s s❡rã♦ ♦ ❝❡♥tr♦ ❝♦♠ ✉♠ ❞❡s❧♦❝❛♠❡♥t♦ ❞❡ c✉♥✐❞❛❞❡s ♣❛r❛ ❞✐r❡✐t❛ ❡ ♦✉tr♦ ♣❛r❛ ❡sq✉❡r❞❛ ♦✉ s❡❥❛✿

F = (m, n)±(c,0). ✭✷✳✼✮

❙❡ ♦ ❡✐①♦ ♠❛✐♦r ❢♦r ♦ ✈❡rt✐❝❛❧ ❡♥tã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦s ❢♦❝♦s s❡rã♦ ♦ ❝❡♥tr♦ ❝♦♠ ✉♠ ❞❡s❧♦❝❛♠❡♥t♦ ❞❡ c ✉♥✐❞❛❞❡s ♣❛r❛ ❝✐♠❛ ❡ ♦✉tr♦ ♣❛r❛ ❜❛✐①♦ ♦✉ s❡❥❛✿

F = (m, n)±(0, c). ✭✷✳✽✮

P❛r❛ r❡♣r❡s❡♥t❛r ❡st❡s ❡❧❡♠❡♥t♦s ✈❛♠♦s ❛♦ ❝❛♠♣♦ ❞❡ ❊♥tr❛❞❛ ❞♦ ●❡♦●❡❜r❛ ❡ ❡♥tr❛✲

r❡♠♦s ❝♦♠ ♦s s❡❣✉✐♥t❡s ❝♦♠❛♥❞♦s

❋❴✶ ❂ ✭♠ ✲ sqrt✭❛❫✷ ✲ ❜❫✷✮✱ ♥✮ ❋❴✷ ❂ ✭♠ ✰ sqrt✭❛❫✷ ✲ ❜❫✷✮✱ ♥✮ ❋❴④✵✶⑥ ❂ ✭♠✱ ♥ ✲ sqrt✭❜❫✷ ✲ ❛❫✷✮✮ ❋❴④✵✷⑥ ❂ ✭♠✱ ♥ ✰ sqrt✭❜❫✷ ✲ ❛❫✷✮✮

❆ss✐♠ t❡♠♦s q✉❛tr♦ ♣♦♥t♦s ❝r✐❛❞♦s✱ ♠❛s ❞♦✐s ❡st❛rã♦ ✐♥❞❡✜♥✐❞♦s ♣♦✐s ♦ ✈❛❧♦r ❞❡♥tr♦ ❞❛ r❛✐③ q✉❛❞r❛❞❛ ❡stá ♥❡❣❛t✐✈♦✱ F1 ❡ F2 sã♦ ♦s ❢♦❝♦s q✉❛♥❞♦ ♦ ❡✐①♦ ❤♦r✐③♦♥t❛❧ é ♠❛✐♦r✱

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✸✽ ❈ô♥✐❝❛s

❋❛③❡♥❞♦ ♥♦✈❛♠❡♥t❡ ❛s ✈❛r✐❛çõ❡s ♥♦s ✈❛❧♦r❡s a✱ b✱ m ❡ n ♣♦❞❡♠♦s ✈❡r✐✜❝❛r s❡ ♥ã♦ ✜③❡♠♦s ♥❡♥❤✉♠ ❡rr♦ ❡ s❡ ♦s ♣♦♥t♦s s❛t✐s❢❛③❡♠ ❛s ❡q✉❛çõ❡s✳

❖✉tr❛ ♣r♦♣r✐❡❞❛❞❡ ❢á❝✐❧ ❞❡ s❡ ♠♦str❛r é ♦ ✈❛❧♦r ❝♦♥st❛♥t❡ ❞❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❞❡ ✉♠ ♣♦♥t♦ ❞❛ ❝✉r✈❛ ❛♦s ❢♦❝♦s✱ ♣r♦♣r✐❡❞❛❞❡ q✉❡ ❞❡✜♥❡ ❛ ❢♦r♠❛ ❣❡♦♠étr✐❝❛✳ P❛r❛ ✐ss♦ ✈❛♠♦s ❝❧✐❝❛r ♥♦ ❜♦tã♦ P♦♥t♦ ❞❡♣♦✐s ❝❧✐❝❛♠♦s ❡♠ ❝✐♠❛ ❞❛ ❡❧✐♣s❡ ♣❛r❛ ❣❡r❛r ✉♠ ♣♦♥t♦✱

♣r♦✈❛✈❡❧♠❡♥t❡ ❝♦♠ ♦ ♥♦♠❡ A✱ s♦❜r❡ ❛ ❡❧✐♣s❡✳ ❆♦ ♠♦✈❡r ♦ ♣♦♥t♦ ❣❡r❛❞♦ ✈❡r❡♠♦s q✉❡ ❡st❡ ♣♦♥t♦ s♦♠❡♥t❡ s❡ ❞❡s❧♦❝❛ ♥❛ tr❛❥❡tór✐❛ ❞♦ ❝♦♥t♦r♥♦ ❞❛ ✜❣✉r❛✳

❋✐❣✉r❛ ✷✳✶✾✿ ❇♦tã♦ q✉❡ ❣❡r❛ ♣♦♥t♦s ♥♦ ♣❧❛♥♦✳

❆♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞♦ ❜♦tã♦ P♦♥t♦ ❤á ♦ ❜♦tã♦ ❘❡t❛ ❝❧✐q✉❡ ♥♦ ♣❡q✉❡♥♦ tr✐â♥❣✉❧♦ ❧♦❝❛❧✐✲ ③❛❞♦ ♥♦ ❝❛♥t♦ ✐♥❢❡r✐♦r ❞✐r❡✐t♦ ❞♦ ❜♦tã♦ ❡ ♠❛✐s ♦♣çõ❡s s✉r❣✐rã♦✳ Pr♦❝✉r❡ ♣♦r ❙❡❣♠❡♥t♦✳

❋✐❣✉r❛ ✷✳✷✵✿ ❊s❝♦❧❤❡♥❞♦ ❛ ❢✉♥çã♦ s❡❣♠❡♥t♦✳

❆♦ ❝❧✐❝❛r♠♦s ❡♠ ✉♠ ❞♦s ❢♦❝♦s ❡ ❞❡♣♦✐s ❡♠ ✉♠ ♣♦♥t♦ s♦❜r❡ ❛ ❡❧✐♣s❡✱ ♦ ♣r♦❣r❛♠❛ ✐rá ❣❡r❛r ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ F1A✳ ❘❡♣✐t❛ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ❣❡r❛r ♦ s❡❣♠❡♥t♦ F2A

❡ ❡♥tã♦ ✈❛♠♦s ♠✉❞❛r ❛s ❞✐♠❡♥sõ❡s ❞❛ ❡❧✐♣s❡ ♣❛r❛ ♣♦❞❡r ❢❛③❡r ❛ ♠❡s♠❛ ❝♦✐s❛ ❝♦♠ ♦s ♣♦♥t♦s F01 ❡ F02✳

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