Programa de Mestrado Pro ssional em Matemática em Rede Nacional

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛

■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠

▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧

❚r❛♥s❧❛çã♦ ❡ r♦t❛çã♦

❞❡ ❝ô♥✐❝❛s ❡♠

R

2

▼❛r❝✐♦ ▲♦♣❡s ❈❛♠♣♦❧✐♥♦

❇r❛sí❧✐❛

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❚r❛♥s❧❛çã♦ ❡ r♦t❛çã♦ ❞❡ ❝ô♥✐❝❛s ❡♠

R

2

♣♦r

▼❛r❝✐♦ ▲♦♣❡s ❈❛♠♣♦❧✐♥♦

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡

▼❊❙❚❘❊ ❊▼ ▼❆❚❊▼➪❚■❈❆

❇r❛sí❧✐❛✱ ✷✹ ❞❡ ❏✉♥❤♦ ❞❡ ✷✵✶✹

❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛✿

❉r✳ ❘✐❝❛r❞♦ ❘✉✈✐❛r♦ ✲ ❯♥❇ ✲ ❖r✐❡♥t❛❞♦r

❉r✳ ❑❡❧❧❝✐♦ ❖❧✐✈❡✐r❛ ❆r❛ú❥♦ ✲ ❯♥❇ ✲ ❊①❛♠✐♥❛❞♦r

❉r✳ ❏♦ã♦ P❛❜❧♦ P✐♥❤❡✐r♦ ❞❛ ❙✐❧✈❛ ✲ ❯❋P❆ ✲ ❊①❛♠✐♥❛❞♦r

❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ❉❡✉s q✉❡✱ s❡❣✉♥❞♦ ●❛❧✐❧❡✉ ●❛❧✐❧❡✐✱ ✉t✐❧✐③♦✉✲s❡ ❞❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ♣❛r❛ ❝♦♥str✉✐r ♦ ♠✉♥❞♦✳

❆ ❯♥❇ ♣♦r t❡r ♠❡ ❛❝♦❧❤✐❞♦ ❞✉r❛♥t❡ ❣r❛❞✉❛çã♦ ❡ ♣ós ❣r❛❞✉❛çã♦✱ ❜❡♠ ❝♦♠♦ s❡✉ ❝♦r♣♦ ❞♦❝❡♥t❡ ❝♦♠♣♦st♦ ♣♦r ♣r♦✜ss✐♦♥❛✐s ❜r✐❧❤❛♥t❡s q✉❡ ❞❡s♣❡rt❛r❛♠ ❡♠ ♠✐♠ ♦ ♣r❛③❡r ♣❡❧❛ ▼❛t❡♠át✐❝❛✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♦ ♣r♦❢❡ss♦r ❉r✳ ❘✐❝❛r❞♦ ❘✉✈✐❛r♦✱ ♣♦r t❡r ❞❡❞✐❝❛❞♦ ❣r❛♥❞❡ ♣❛rt❡ ❞♦ s❡✉ t❡♠♣♦ à ♠✐♥❤❛ ❝❛✉s❛✳ P♦r t❡r s✐❞♦ ✉♠ ♣❛r❝❡✐r♦ ❡ ❛♣♦✐❛❞♦r ❞❡st❡ tr❛❜❛❧❤♦✳

❆♦s ♠❡✉s ♣❛✐s✱ ❘❡♥❛t♦ ❡ ❚❡❧♠❛✱ ♣♦r t❡r❡♠ ♠❡ ❞❛❞♦ ❝♦♥❞✐çõ❡s ❞❡ ❝❤❡❣❛r ❛té ❛q✉✐✳ P❡❧♦ ❛♠♦r q✉❡ s❡♠♣r❡ ❞❡♠♦♥str❛r❛♠ ❡ ♣♦r t✉❞♦ q✉❡ ❡❧❡s ❢❛③❡♠ ♣♦r ♠✐♠✳

❆ ♠✐♥❤❛ ❡s♣♦s❛✱ ▲♦✉❛♥❛✱ ❝♦♠♣❛♥❤❡✐r❛ ❛♠á✈❡❧ q✉❡ ❉❡✉s ❡s❝♦❧❤❡✉ ♣❛r❛ ❡st❛r s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦✳

❆♦ ❚❤❡♦✱ ♠❡✉ ♣r✐♠❡✐r♦ ✜❧❤♦ q✉❡ ❡stá ♣♦r ✈✐r✱ q✉❡ ♠❡ ✐♥❝❡♥t✐✈♦✉ ❛ ♥ã♦ ❞❡s✐st✐r ❡ ♥❡♠ ♠❡s♠♦ ❞❡s❛♥✐♠❛r ❝♦♠ ♦ ❝❛♥s❛ç♦✳

❘❡s✉♠♦

❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ✐❞❡♥t✐✜❝❛r ❛ ❝ô♥✐❝❛ r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ ✐♥✐❝✐❛❧♠❡♥t❡ ❢♦r❛♠ ❛♣r❡s❡♥t❛❞❛s ❛s ❡q✉❛çõ❡s ❝❛♥ô♥✐❝❛s ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❡❧✐♣s❡✱ ❤✐♣ér❜♦❧❡ ❡ ♣❛rá❜♦❧❛✳ ❊♠ s❡❣✉✐❞❛ ✈❡r✐✜❝♦✉✲s❡ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ s✐♠♣❧✐✜❝❛r ❛ ❡s❝r✐t❛ ❞❡ ❛❧❣✉♠❛s ❡q✉❛çõ❡s✱ ❛ ✜♠ ❞❡ ✐❞❡♥t✐✜❝❛r ❛ ❝ô♥✐❝❛ ❡ s❡✉s ♣r✐♥❝✐♣❛✐s ❡❧❡♠❡♥t♦s✳ ❊♥tr❡✲ t❛♥t♦✱ ❢♦✐ ♥❡❝❡ssár✐♦ ✉♠ ❧❡✈❛♥t❛♠❡♥t♦ t❡ór✐❝♦ ❛❝❡r❝❛ ❞♦s ✈❡t♦r❡s ❡ ❞❡ s✉❛ ❛♣❧✐❝❛çã♦ ♥❛ tr❛♥s❧❛çã♦ ❡ r♦t❛çã♦ ❞❡ ♣♦♥t♦s ❡ ❝✉r✈❛s ❡♠ ✉♠ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✳

P♦r ✜♠✱ ❢♦✐ ✈✐st♦ ❝♦♠♦ ❡❧✐♠✐♥❛r ♦s t❡r♠♦s ❧✐♥❡❛r❡s ❡ ♦ t❡r♠♦ q✉❛❞rát✐❝♦ ♠✐st♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ t♦r♥❛♥❞♦ ❛ ❡q✉❛çã♦ ♠❛✐s s✐♠♣❧❡s ❡ ❛ ✐❞❡♥t✐✜❝❛✲ çã♦ ❞❛ ❝ô♥✐❝❛ ❝♦♠♦ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❡❧✐♣s❡✱ ❤✐♣ér❜♦❧❡ ♦✉ ♣❛rá❜♦❧❛✱ ❜❡♠ ❝♦♠♦ ❞❡ s❡✉s ♣r✐♥❝✐♣❛✐s ❡❧❡♠❡♥t♦s✱ ✉♠❛ t❛r❡❢❛ ♠❛✐s ❢á❝✐❧✳

P❛❧❛✈r❛s✲❝❤❛✈❡ ❈ô♥✐❝❛s❀ ◗✉á❞r✐❝❛s❀ ❊q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉❀ ❘♦t❛çã♦❀ ❚r❛♥s❧❛çã♦❀ ❱❡t♦r❡s✳

❆❜str❛❝t

❆✐♠✐♥❣ t♦ ✐❞❡♥t✐❢② t❤❡ ❝♦♥✐❝ r❡♣r❡s❡♥t❡❞ ❜② ❛ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥✱ ✐♥✐t✐❛❧❧② t❤❡ ❝❛✲ ♥♦♥✐❝❛❧ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❝✐r❝❧❡✱ ❡❧❧✐♣s❡✱ ♣❛r❛❜♦❧❛ ❛♥❞ ❤②♣❡r❜♦❧❛ ✇❡r❡ ♣r❡s❡♥t❡❞✳ ❚❤❡♥ t❤❡r❡ ✐s t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ s✐♠♣❧✐❢②✐♥❣ t❤❡ ✇r✐t✐♥❣ ♦❢ s♦♠❡ ❡q✉❛t✐♦♥s ✐♥ ♦r❞❡r t♦ ✐❞❡♥✲ t✐❢② t❤❡ ❝♦♥✐❝❛❧ ❛♥❞ ✐ts ♠❛✐♥ ❡❧❡♠❡♥ts✳ ❍♦✇❡✈❡r✱ ✇❡ ♥❡❡❞❡❞ ❛ t❤❡♦r❡t✐❝❛❧ s✉r✈❡② ♦♥ t❤❡ ✈❡❝t♦rs ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s ✐♥ tr❛♥s❧❛t✐♦♥ ❛♥❞ r♦t❛t✐♦♥ ♦❢ t❤❡ ❝✉r✈❡s ❛♥❞ ♣♦✐♥ts ✐♥ ❛ ❈❛rt❡s✐❛♥ ♣❧❛♥❡✳

❋✐♥❛❧❧②✱ ✐t ✇❛s s❡❡♥ ❛s ❡❧✐♠✐♥❛t❡ t❤❡ ❧✐♥❡❛r t❡r♠s ❛♥❞ t❤❡ q✉❛❞r❛t✐❝ ♠✐①❡❞ t❡r♠ ♦❢ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✱ ♠❛❦✐♥❣ t❤❡ s✐♠♣❧❡st ❡q✉❛t✐♦♥ ❛♥❞ t❤❡ ✐❞❡♥t✐❢② ♦❢ t❤❡ ❝♦♥✐❝ ❛s ❝✐r❝❧❡✱ ❡❧❧✐♣s❡✱ ❤②♣❡r❜♦❧❛✱ ♦r ♣❛r❛❜♦❧❛✱ ❛s ✇❡❧❧ ❛s ✐ts ♠❛✐♥ ❝♦♠♣♦♥❡♥ts✱ ❛♥ ❡❛s✐❡r t❛s❦✳

❑❡②✇♦r❞s ❈♦♥✐❝s❀ ◗✉❛❞r✐❝s❀ ◗✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s❀ ❘♦t❛t✐♦♥❀ ❚r❛♥s❧❛t✐♦♥❀ ❱❡❝t♦rs✳

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶ ❉✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❡♠ ✉♠ ♣❧❛♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ C ❡ r❛✐♦ r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✸ ❊❧✐♣s❡ ❝ê♥tr✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✹ ❈❛s♦s ❡s♣❡❝✐❛✐s ❞❡ ❡❧✐♣s❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✺ ❊❧✐♣s❡ ❝♦♠ ❢♦❝♦s A= (1,4)❡ B = (9,4)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✻ ❊❧✐♣s❡ ❝♦♠ ❢♦❝♦s A= (₋2√6,₋2√6)❡ B = (2√6,2√6)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✼ ❍✐♣ér❜♦❧❡ ❝ê♥tr✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✽ ❍✐♣ér❜♦❧❡ ❝♦♠ ❢♦❝♦s A= (2,5)❡ B = (₋2,3)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✾ P❛rá❜♦❧❛ ❝♦♠ ✈ért✐❝❡ (0,0)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✶✵ P❛rá❜♦❧❛ ❝♦♠ r❡t❛ ❞✐r❡tr✐③r :x+ 2y= 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✶✶ ❙❡❣♠❡♥t♦s ❝♦❧✐♥❡❛r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✷ ❙❡❣♠❡♥t♦s ♣❛r❛❧❡❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✸ ❚r❛♥s❧❛çã♦ ❞❛ ♣❛rá❜♦❧❛ y=x2

+ 3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✶✹ ❚r❛♥s❧❛çã♦ ❞❛ ❡❧✐♣s❡ 9x2

+ 25y2

−90x₋200y+ 400 = 0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✶✺ ❆❞✐çã♦ ❞❡ ✈❡t♦r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶✻ ➶♥❣✉❧♦ ❡♥tr❡ ✈❡t♦r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✶✼ Pr♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞❡~u s♦❜r❡ ~v✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✶✽ ❘♦t❛çã♦ ❞❡ ✉♠ â♥❣✉❧♦ θ ♥♦ s❡♥t✐❞♦ ❛♥t✐✲❤♦rár✐♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✶✾ ❚r❛♥s❧❛çã♦ ❡ r♦t❛çã♦ ❞❛ ❤✐♣ér❜♦❧❡ x2

−6xy+y2

+ 2x₋8y₋4 = 0✳ ✳ ✳ ✹✼

✷✵ ❚r❛♥s❧❛çã♦ ❡ r♦t❛çã♦ ❞❛ ❡❧✐♣s❡ 3x2

+ 4xy+ 4y2

−2x−4y−1 = 0✳ ✳ ✳ ✳ ✹✾

✷✶ ❘♦t❛çã♦ ❡ tr❛♥s❧❛çã♦ ❞❛ ♣❛rá❜♦❧❛4x2

−4xy+y2

−18x₋16y+ 39 = 0✳ ✺✷

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✼

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

✷ ❱❡t♦r❡s ♥♦ ♣❧❛♥♦ ❡ tr❛♥s❧❛çã♦ ✷✹ ✷✳✶ ❚r❛♥s❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷ ❖♣❡r❛çõ❡s ❝♦♠ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷✳✶ ❆❞✐çã♦ ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠ ✈❡t♦r ♣♦r ✉♠ ♥ú♠❡r♦ r❡❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✷✳✸ Pr♦❞✉t♦ ✐♥t❡r♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✷✳✹ Pr♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞❡ ✉♠ ✈❡t♦r s♦❜r❡ ♦✉tr♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✸ ❘♦t❛çã♦ ❡ ♠✉❞❛♥ç❛ ❞❡ ❡✐①♦s ✸✼ ✸✳✶ ▼✉❞❛♥ç❛ ❞❡ ❡✐①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✷ ❋♦r♠❛s ◗✉á❞r✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✸ ❊q✉❛çã♦ ❣❡r❛❧ ❞♦ ✷♦ _{❣r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺}

✹ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✺✸

■♥tr♦❞✉çã♦

❖ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦ ♥♦ ❇r❛s✐❧ é ❜❛st❛♥t❡ ❣❡♥❡r❛❧✐st❛✱ ❛❜♦r✲ ❞❛♥❞♦ ✈ár✐❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛ ❝♦♠ ❞✐❢❡r❡♥t❡s ♥í✈❡✐s ❞❡ ❛♣r♦❢✉♥❞❛♠❡♥t♦✳ ❯♠ ❞❡❧❡s ♠❡ ❝❤❛♠♦✉ ❜❛st❛♥t❡ ❛t❡♥çã♦ ❞✉r❛♥t❡ ❛ ♠✐♥❤❛ ♣❛ss❛❣❡♠ ♣❡❧♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♦ ❡st✉❞♦ ❞❡ ❡q✉❛çõ❡s ❡ ❣rá✜❝♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞❛s ❝ô♥✐❝❛s✳ ❈♦♥t❡ú❞♦ q✉❡ ❞✐✈✐❞❡ ♦♣♥✐õ❡s ❛té ♠❡s♠♦ ❡♥tr❡ ♦s ♠❛t❡♠át✐❝♦s✳ ❆♠❛❞❛ ♣♦r ♠✉✐t♦s ❡ ♦❞✐❛❞❛ ♣♦r t❛♥t♦s ♦✉tr♦s✳

❈♦♥t❡ú❞♦ ❜❛st❛♥t❡ r✐❝♦ ❡♠ ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ tr❛❜❛❧❤♦ ❡ ❝♦♠ ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s ♥❛s ❡♥❣❡♥❤❛r✐❛s ❡ ❡♠ ♦✉tr❛s ár❡❛s✱ ♦ ❡st✉❞♦ ❞❛s ❝ô♥✐❝❛s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ r❡s✉♠❡✲s❡ ❛♦ ❡st✉❞♦ ❞❡ ❝ô♥✐❝❛s ❡♠ q✉❡ ♦ ❡✐①♦ ❞❡ s✐♠❡tr✐❛✱ ♦✉ r❡t❛ ❢♦❝❛❧✱ ❡stá s❡♠♣r❡✱ ♦✉ q✉❛s❡ s❡♠♣r❡✱ ♣❛r❛❧❡❧♦ ❛ ✉♠ ❞♦s ❡✐①♦s ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛❞♦t❛❞♦✳

❊♥tr❡t❛♥t♦✱ q✉❡st✐♦♥❛♠❡♥t♦ s♦❜r❡ ❝♦♠♦ s❡r✐❛♠ ❛s ❡q✉❛çõ❡s ❞❡ ❝ô♥✐❝❛s ❡♠ ♦✉tr❛s ♣♦s✐çõ❡s ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ♣♦❞❡♠ s✉r❣✐r ♣♦r ♣❛rt❡ ❞♦s ❛❧✉♥♦s✳ ◗✉❛♥❞♦ ✐ss♦ ❛❝♦♥t❡❝❡✱ ♠✉✐t♦s ♣r♦❢❡ss♦r❡s ❡sq✉✐✈❛♠✲s❡✱ ♠✉✐t❛s ✈❡③❡s ♣♦r ♥ã♦ t❡r❡♠ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ s✉✜❝✐❡♥t❡ ❛ ♣♦♥t♦ ❞❡ s❡ s❡♥t✐r❡♠ s❡❣✉r♦s ❡♠ tr❛❜❛❧❤❛r ❝♦♠ ❡❧❛s ❡♠ s❛❧❛✳

❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❞❛r à ❡ss❡s ♣r♦❢❡ss♦r❡s✱ ❜❡♠ ❝♦♠♦ ❛♦s ❛❧✉♥♦s✱ ❜❛s❡ ♣❛r❛ ❡st✉❞❛✲ r❡♠ ❛s ❝ô♥✐❝❛s ❡♠ q✉❡ ❛ r❡t❛ ❢♦❝❛❧ ♥ã♦ é ♣❛r❛❧❡❧❛ ❛ ♥❡♥❤✉♠ ❞♦s ❡✐①♦s✱ ❢♦✐ q✉❡ ❞❡❝✐❞✐ ❡s❝r❡✈❡r s♦❜r❡ ❡ss❡ t❡♠❛✳

▼♦str❛r❡✐ ♥♦ ♠❡✉ tr❛❜❛❧❤♦ q✉❡✱ ❝♦♠ ✉♠ ♣♦✉❝♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ s♦❜r❡ ❛s ❡q✉❛çõ❡s ❝❛♥ô♥✐❝❛s ❞❛s ❝ô♥✐❝❛s✱ ✈❡t♦r❡s ❡ s✉❛s ♦♣❡r❛çõ❡s✱ ❡ ♠❛tr✐③❡s ✭❝♦♥t❡ú❞♦s tr❛❜❛❧❤❛❞♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✮✱ é ♣♦ssí✈❡❧ tr❛♥s❢♦r♠❛r♠♦s ❛s ❡q✉❛çõ❡s ❞❡ss❛s ❝ô♥✐❝❛s ❡ r❡❡s❝r❡✈ê✲❧❛s ♥❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛✱ ❢❛❝✐❧✐t❛♥❞♦ ❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ q✉❛❧ é ❛ ❝ô♥✐❝❛ r❡♣r❡s❡♥t❛❞❛ ❡ q✉❛✐s sã♦ s❡✉s ♣r✐♥❝✐♣❛✐s ❡❧❡♠❡♥t♦s✳

◆❛ ♣r✐♠❡✐r❛ s❡çã♦ ❞♦ tr❛❜❛❧❤♦✱ ❢♦r❛♠ ❛❜♦r❞❛❞❛s ❛s ❞❡✜♥✐çõ❡s ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❝ô♥✐❝❛s tr❛t❛♥❞♦✲❛s ❝♦♠♦ ❧✉❣❛r❡s ❣❡♦♠étr✐❝♦s✳ ❆ ♣❛rt✐r ❞❡ss❛s ❞❡✜♥✐çõ❡s ❢♦r❛♠ ♦❜t✐❞❛s ❛s ❡q✉❛çõ❡s ❝❛♥ô♥✐❝❛s ❞❡ ❝❛❞❛ ❝ô♥✐❝❛✳

❋❛③❡♥❞♦ ✉♠❛ r❡✢❡①ã♦ s♦❜r❡ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ✈❡t♦r❡s ♥❛ tr❛♥s❧❛çã♦ ❞❡ ♣♦♥t♦s ❡ ❝✉r✈❛s ❡♠ ✉♠ ♣❧❛♥♦✱ ♥❛ s❡❣✉♥❞❛ s❡çã♦ ❞♦ tr❛❜❛❧❤♦ ❢♦✐ ❞❡✜♥✐❞♦ ♦ q✉❡ é ✉♠ ✈❡t♦r ❡ ❛♣r❡s❡♥t❛❞❛s ❛❧❣✉♠❛s ♦♣❡r❛çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s q✉❡✱ ♣❛r❛ ♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✱ ❡r❛♠ ♣ré✲r❡q✉✐s✐t♦s✳ ❆✐♥❞❛ ♥❡ss❛ s❡çã♦✱ ✈✐♠♦s ❝♦♠♦ tr❛♥s❧❛❞❛r ✉♠ ❝✉r✈❛ ❡ ♦❜t❡r s✉❛ ♥♦✈❛ ❡q✉❛çã♦✳

❈♦♠♦ ♣❛rt❡ ❞♦ ♦❜❥❡t✐✈♦✱ ❛ tr❛♥s❧❛çã♦ ❢♦✐ ✉t✐❧✐③❛❞❛ ❝♦♠♦ ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ s✐♠✲ ♣❧✐✜❝❛r ❛ ❡s❝r✐t❛ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❞❛ ❢♦r♠❛Ax2

+Bxy+Cy2

+

✱ ❡❧✐♠✐♥❛♥❞♦✲s❡ ♦s t❡r♠♦s ❧✐♥❡❛r❡s✳

◆❛ t❡r❝❡✐r❛ ❡ ú❧t✐♠❛ s❡çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✱ ❢♦✐ ❛♣r❡s❡♥t❛❞❛ ❛ ♠✉❞❛♥ç❛ ❞❡ ❡✐①♦s ♣♦r ♠❡✐♦ ❞❛ r♦t❛çã♦ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡ ❝♦♠♦✱ ♣♦r ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ r♦t❛çã♦✱ ❡❧✐♠✐♥❛r ♦ t❡r♠♦ q✉❛❞rát✐❝♦ ♠✐st♦✱Bxy✱ t♦r♥❛♥❞♦ ❛ ❡q✉❛çã♦ ♠❛✐s s✐♠♣❧❡s ❡ ❛ ❝ô♥✐❝❛

❝♦♠ r❡t❛ ❢♦❝❛❧ ♥❛ ♠❡s♠❛ ❞✐r❡çã♦ ❞❡ ✉♠ ❞♦s ❡✐①♦s ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛❞♦t❛❞♦✳

✶ ❈ô♥✐❝❛s ❡ ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞♦ s❡❣✉♥❞♦ ❣r❛✉

❯♠❛ ❝ô♥✐❝❛ ♣♦❞❡ s❡r ♦❜t✐❞❛ ♣❡❧❛ ✐♥t❡rs❡❝çã♦ ❞❡ ✉♠ ♣❧❛♥♦ ❝♦♠ ✉♠❛ s✉♣❡r❢í❝✐❡ ❝ô♥✐❝❛ ❞❡ r❡✈♦❧✉çã♦ ❡ t❡♠ ♣♦r ❡q✉❛çã♦✱ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✱ ✉♠❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ ♣♦❞❡♥❞♦ s❡r ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ✉♠❛ ❡❧✐♣s❡✱ ✉♠❛ ❤✐♣ér❜♦❧❡ ♦✉ ✉♠❛ ♣❛rá❜♦❧❛✳

◆❡ss❛ ♣r✐♠❡✐r❛ s❡çã♦ ❞♦ tr❛❜❛❧❤♦ r❡❝♦r❞❛r❡♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ❝❛❞❛ ✉♠❛ ❞❡ss❛s ❝ô♥✐❝❛s ❛♥❛❧✐s❛♥❞♦✲❛s ❝♦♠♦ ❧✉❣❛r❡s ❣❡♦♠étr✐❝♦s✱ ✐st♦ é✱ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s q✉❡ ♣♦ss✉❡♠ ✉♠❛ ♠❡s♠❛ ♣r♦♣r✐❡❞❛❞❡✳

✶✳✶ ❈✐r❝✉♥❢❡rê♥❝✐❛

❆♥t❡s ❞❡ ❢❛❧❛r♠♦s s♦❜r❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ✈❛❧❡ ❧❡♠❜r❛r q✉❡✱ ❞❛❞♦s ❞♦✐s ♣♦♥t♦s

A = (a1, a2) ❡ B = (b1, b2)✱ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡❧❡s é ✐❣✉❛❧ ❛ ♠❡❞✐❞❛ ❞♦ s❡❣✉✐♠❡♥t♦

❞❡ r❡t❛AB✱ ✐♥❞✐❝❛❞❛ ♣♦r d(A, B)✳

A

B

O

_x

y

a

a

b

b

b

-a

b

-a

❋✐❣✉r❛ ✶✿ ❉✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❡♠ ✉♠ ♣❧❛♥♦✳

P❡❧♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ d(A, B)2

= (b1 ₋a1)2

+ (b2₋a2)2✳ ❖ q✉❡ ✐♠♣❧✐❝❛ ❡♠ d(A, B) =p

(b1 ₋a1)2_{+ (b2−a2)}2✳

❉❡✜♥✐çã♦ ✶✳ ❈✐r❝✉♥❢❡rê♥❝✐❛ é ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ❞❡ ✉♠ ♣❧❛♥♦ ❡q✉✐❞✐st❛♥t❡s ❞❡ ✉♠ ♣♦♥t♦ ✜①♦✱ ❞❡♥♦♠✐♥❛❞♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳

P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❞❛❞♦s ✱ ❝❡♥tr♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❡r ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦✱ ♣❛r❛ t♦❞♦ ♣♦♥t♦P = (x, y)✱ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ t❡♠✲s❡ q✉❡

d(P, C) =r✳ ❉❛ ❞❡✜♥✐çã♦✱ s❡❣✉❡ q✉❡✿

d(P, C) = r

⇔p(x−x0)2_{+ (y−y0)}2 _=r

⇔(x−x0)2

+ (y−y0)2

=r2

. ✭✶✮

C

P r

❋✐❣✉r❛ ✷✿ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ C ❡ r❛✐♦ r✳

❉✐③❡♠♦s q✉❡ ✭✶✮ é ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝✉❥♦ ❝❡♥tr♦ é ♦ ♣♦♥t♦(x0, y0)

❡r ♦ r❛✐♦ ❞❡ss❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳

❊①❡♠♣❧♦ ✶✳ ❉❡t❡r♠✐♥❛r ❛ ❡q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ 5✱ ❝❡♥tr❛❞❛ ♥♦ ♣♦♥t♦

C= (3,2)✳

❙♦❧✉çã♦✿ P❡❧❛ ❞❡✜♥✐çã♦✱ s❡ P = (x, y) é ✉♠ ♣♦♥t♦ ❞❡ss❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❡♥tã♦

d(P, C) =r✳ ▲♦❣♦✿ p

(x−3)2_{+ (y−2)}2 _{= 5 ⇔ (x−3)}2

+ (y−2)2

= 52 .

❊ss❛ é ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❡♥tr❡t❛♥t♦✱ s❡ ❛ ❡s❝r❡✈❡r♠♦s ❞❡s❡♥✈♦❧✲ ✈❡♥❞♦ ❛s ♣♦tê♥❝✐❛s✱ t❡r❡♠♦s✿

x2

−6x+ 9 +y2

−4y+ 4 = 25

⇔x2

+y2

−6x−4y−12 = 0. ✭✷✮

❉✐③❡♠♦s q✉❡ ✭✷✮ é ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ C = (3,2) ❡ r❛✐♦

r= 5✳ ✷

❊①❡♠♣❧♦ ✷✳ ❙❡❥❛x2

+y2

−10x+ 2y+ 10 = 0 ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳

❉❡t❡r♠✐♥❡ ♦ ❝❡♥tr♦ ❡ ♦ r❛✐♦ ❞❡ss❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳

❙♦❧✉çã♦✿ P❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ♦ ❝❡♥tr♦ ❡ ♦ r❛✐♦ ❞❡ss❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❜❛st❛ ❡s❝r❡✈❡r ❛ s✉❛ ❡q✉❛çã♦ ♥❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛✳ ◆♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ♣❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❛q✉❡❧❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❞❡s❡♥✈♦❧✈❡♠♦s ❛s ♣♦tê♥❝✐❛s ❡ r❡❞✉③✐♠♦s ♦s t❡r♠♦s s❡♠❡✲ ❧❤❛♥t❡s✳ ❆❣♦r❛ ❢❛r❡♠♦s ♦ ❝♦♥trár✐♦✱ ✉t✐❧✐③❛r❡♠♦s ❛ té❝♥✐❝❛ ❞❡ ❝♦♠♣❧❡t❛r q✉❛❞r❛❞♦s ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❡ss❛✳

x2

+y2

−10x+ 2y+ 10 = 0

⇔x2

−10x+ 25 +y2

+ 2y+ 1 =₋10 + 25 + 1

⇔(x−5)2

+ (y+ 1)2

= 16

⇔(x−5)2

+ (y−(−1))2

= 42 .

◆❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛(x₋x0)2

+ (y₋y0)2

=r2✱

(x0, y0)é ♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛

❡r ♦ r❛✐♦✳ ▲♦❣♦✱ (5,₋1) é ♦ ❝❡♥tr♦ ❞❡ss❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ r= 4 ♦ s❡✉ r❛✐♦✳ ✷

✶✳✷ ❊❧✐♣s❡

◆❛ s❡q✉ê♥❝✐❛✱ ❢❛❧❛r❡♠♦s s♦❜r❡ ❛ ❡❧✐♣s❡✱ ✉♠❛ ✐♠♣♦rt❛♥t❡ ❝✉r✈❛ ❝✉❥❛s ❛♣❧✐❝❛çõ❡s ✈ã♦ ❞❡s❞❡ ♦ s✐♠♣❧❡s ❡st✉❞♦ ❞❡ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛ ❛ ❝♦♥t❡①t♦s ❜❡♠ ♠❛✐s ❛✈❛♥ç❛❞♦s✱ ❝♦♠♦ ♦ ❡st✉❞♦ ❞♦ ♠♦✈✐♠❡♥t♦ ❞♦s ❛str♦s✳

❉❡✜♥✐çã♦ ✷✳ ❊❧✐♣s❡ é ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ❞❡ ✉♠ ♣❧❛♥♦ ❝✉❥❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❛ ❞♦✐s ♣♦♥t♦s ✜①♦s ✭❢♦❝♦s✮ é ❝♦♥st❛♥t❡ ❡ ♠❛✐♦r ❞♦ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❢♦❝♦s✳

❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ❝♦♥s✐❞❡r❡♠♦s ✉♠❛ ❡❧✐♣s❡ ❝❡♥tr❛❞❛ ♥❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ❝✉❥♦s ❢♦❝♦sA❡B ❡stã♦ ❧♦❝❛❧✐③❛❞♦s s♦❜r❡ ♦ ❡✐①♦Ox✱ ❛ss✐♠A= (−c,0)

❡B = (c,0)✳ ❆ r❡t❛ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦s ❢♦❝♦s ❞❛ ❡❧✐♣s❡ é ❞❡♥♦♠✐♥❛❞❛ r❡t❛ ❢♦❝❛❧✳ ❆ r❡t❛

♣❡r♣❡♥❞✐❝✉❧❛r à r❡t❛ ❢♦❝❛❧ ♣❡❧♦ ❝❡♥tr♦ ❞❛ ❡❧✐♣s❡ é ❛ r❡t❛ ♥ã♦ ❢♦❝❛❧✳ ❉❡♥♦t❛r❡♠♦s ♣♦rb

❛ ❞✐stâ♥❝✐❛ ❞♦ ❝❡♥tr♦ ❞❛ ❡❧✐♣s❡ ❛ ❝❛❞❛ ✉♠ ❞♦s ✈ért✐❝❡s ❞❛ ❡❧✐♣s❡ s♦❜r❡ ❛ r❡t❛ ♥ã♦ ❢♦❝❛❧✳

x

A

=(-c,0)

B

=(c,0)

C

=(0,b)

a

a

O

P

=(

x

,

y

)

❋✐❣✉r❛ ✸✿ ❊❧✐♣s❡ ❝ê♥tr✐❝❛✳

P♦r ❞❡✜♥✐çã♦✱ s❡2aé ❛ ❝♦♥st❛♥t❡ q✉❡ r❡♣r❡s❡♥t❛ ❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❞❡ ✉♠ ♣♦♥t♦ P = (x, y)❞❛ ❡❧✐♣s❡ ❛ ❝❛❞❛ ✉♠ ❞♦s ❢♦❝♦s✱ ❡♥tã♦✿

d(P, A) +d(P, B) = 2a

⇔p(x+c)2_{+ (y}

−0)2₊p_(x

−c)2_{+ (y}

−0)2 _{= 2a}

⇔p(x−c)2_{+ (y−0)}2 _{= 2a−}p_(x+c)2_{+ (y−0)}2_. ✭✸✮

❊❧❡✈❛♥❞♦ ❛♠❜♦s ♠❡♠❜r♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ✭✸✮ ❛♦ q✉❛❞r❛❞♦✱ t❡♠♦s✿

⇔x2

−2xc+c2

+y2

= 4a2

+x2

+c2

+ 2xc+y2

−4ap(x+c)2_+y2

⇔a2

+xc=ap(x+c)2_+y2

⇔a4

+ 2a2

xc+x2 c2

=a2 x2

+a2 c2

+ 2a2

xc+a2 y2

⇔x2

(a2

−c2

) +y2 a2

=a2

(a2

−c2

).

▼❛sa2

−c2

=b2_{✱ ♣♦rt❛♥t♦✱} x2

b2

+y2 a2

=a2

b2_{✳ ❉✐✈✐❞✐♥❞♦ ❛ ❡q✉❛çã♦ ♣♦r} a2

b2_{✱ t❡♠♦s✿}

x2 a2 +

y2

b2 = 1. ✭✹✮

❚❡♠✲s❡ q✉❡ ✭✹✮ é ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❛ ❡❧✐♣s❡ ❝❡♥tr❛❞❛ ♥♦ ♣♦♥t♦ O = (0,0)✱ ❝✉❥❛

r❡t❛ ❢♦❝❛❧ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ❡✐①♦Ox✳

P❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ ❛ ❡❧✐♣s❡ ❡stá ❝❡♥tr❛❞❛ ♥❛ ♦r✐❣❡♠ ❡ ❛ r❡t❛ ❢♦❝❛❧ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ❡✐①♦Oy✱ t❡r❡♠♦s ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛✿

x2 b2 +

y2 a2 = 1.

❆♦ ❡st✉❞❛r♠♦s ✉♠❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ✐❞❡♥t✐✜❝❛r q✉❛❧ ❛ ❝ô♥✐❝❛ q✉❡ ❡❧❛ r❡♣r❡s❡♥t❛✳ ▼❛s ♥ã♦ ❜❛st❛ r❡❝♦♥❤❡❝❡r ❛ ❝ô♥✐❝❛ ❝♦♠♦ ✉♠❛ ❡❧✐♣s❡✱ ❤✐♣ér❜♦❧❡ ♦✉ ♣❛rá❜♦❧❛✱ s❡ ❢❛③ ♥❡❝❡ssár✐♦ ✐❞❡♥t✐✜❝❛r ❛❧❣✉♥s ❞❡ s❡✉s ❡❧❡♠❡♥t♦s✳ ◆❛ ❡❧✐♣s❡✱ ♦s ♣r✐♥❝✐♣❛✐s ❡❧❡♠❡♥t♦s ❛ s❡r❡♠ ✐❞❡♥t✐✜❝❛❞♦s sã♦✿ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ❢♦❝♦✱ ❛s ❝♦♦r❞❡✲ ♥❛❞❛s ❞♦ ❝❡♥tr♦ ❞❛ ❡❧✐♣s❡✱ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❢♦❝♦s 2c✱ ❛ r❡t❛ ❢♦❝❛❧✱ ❛ r❡t❛ ♥ã♦ ❢♦❝❛❧

✭♣❡r♣❡♥❞✐❝✉❧❛r à r❡t❛ ❢♦❝❛❧ ♥♦ ❝❡♥tr♦ ❞❛ ❡❧✐♣s❡✮ ❡ ❛ ❡①❝❡♥tr✐❝✐❞❛❞❡e✱ ❞❡✜♥✐❞❛ ♣❡❧❛ r❛③ã♦

❞❡c♣❛r❛ a✱ ✐st♦ é✱e= c

a✳

❙♦❜r❡ ❛ ❡①❝❡♥tr✐❝✐❞❛❞❡✱ ♥♦t❡ q✉❡ 0 _≤ c _≤ a✱ ♣♦rt❛♥t♦ 0 _≤ e _≤ 1✳ ❙❡ ❛♥❛❧✐s❛r♠♦s

♦s ❝❛s♦s ❡①tr❡♠♦s✱ ✐st♦ é✱ s❡ c = 0✱ ♦s ❞♦✐s ❢♦❝♦s ❝♦✐♥❝✐❞❡♠ ❡ t❡♠♦s ♣♦rt❛♥t♦ ✉♠❛

❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ 2a✳ P❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ c = a✱ ❛ ❡❧✐♣s❡ s❡ ❞❡❣❡♥❡r❛ ❡♠ ✉♠

s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❝✉❥♦s ❡①tr❡♠♦s sã♦ ♦s ❢♦❝♦s✳

O x

y

O x

y

O x

y

(a) Elipse. (b) Círculo. (c) Segmento.

❋✐❣✉r❛ ✹✿ ❈❛s♦s ❡s♣❡❝✐❛✐s ❞❡ ❡❧✐♣s❡s✳

❊①❡♠♣❧♦ ✸✳ ❯t✐❧✐③❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❡❧✐♣s❡ ❝♦♠♦ ✉♠ ❧✉❣❛r ❣❡♦♠étr✐❝♦✱ ❞❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ ❡❧✐♣s❡ ❝✉❥♦s ❢♦❝♦s sã♦ ♦s ♣♦♥t♦sA = (1,4)❡B = (9,4)✱ t❛❧ q✉❡ ❛ s♦♠❛ ❞❛s

❞✐stâ♥❝✐❛s ❞❡ ✉♠ ♣♦♥t♦ ❞❛ ❡❧✐♣s❡ ❛ ❝❛❞❛ ✉♠ ❞♦s ❢♦❝♦s é2a= 10✳

❙♦❧✉çã♦✿ ◆♦t❡ q✉❡ ♦s ❢♦❝♦s ❡stã♦ s♦❜r❡ ❛ r❡t❛ y = 4✱ ❡ q✉❡ ♦ ❝❡♥tr♦ ❞❛ ❡❧✐♣s❡ é ♦

♣♦♥t♦ ♠é❞✐♦ ❞♦s ❢♦❝♦s✱ ♦✉ s❡❥❛✱ s❡Cé ♦ ❝❡♥tr♦ ❞❛ ❡❧✐♣s❡ ❞❡ ❢♦❝♦sA = (1,4)❡B = (9,4)✱

❡♥tã♦ C = 1+9 2 ,

4+4 2

= (5,4)✳ ▲♦❣♦✱ ❛ r❡t❛ ♥ã♦ ❢♦❝❛❧ é x= 5✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡

❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❢♦❝♦s é t❛❧ q✉❡2c= 8_⇒c= 4✳ ▼❛s s❡ a= 5✱ c= 4 ❡b2

=c2

−a2_✱

❡♥tã♦ b= 3✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡ ❛ ❡①❝❡♥tr✐❝✐❞❛❞❡ e= c

a =

4

5 = 0,8✳

A B

4

x

1 9

O

❋✐❣✉r❛ ✺✿ ❊❧✐♣s❡ ❝♦♠ ❢♦❝♦s A= (1,4) ❡B = (9,4)✳

❘❡❝♦♥❤❡❝✐❞♦s ❛❧❣✉♥s ❡❧❡♠❡♥t♦s ❞❡ss❛ ❡❧✐♣s❡✱ ✉t✐❧✐③❛r❡♠♦s ♦ ❢❛t♦ ❞❡ q✉❡✱ ♣❡❧❛ ❞❡✲ ✜♥✐çã♦✱ s❡ P = (x, y) é ✉♠ ♣♦♥t♦ ❞❡ss❛ ❡❧✐♣s❡✱ ❡♥tã♦ d(P, A) +d(P, B) = 2a✱ ♣❛r❛

❞❡t❡r♠✐♥❛r♠♦s s✉❛ ❡q✉❛çã♦✿

p

(x−1)2_{+ (y−4)}2₊p_(x−9)2_{+ (y−4)}2 _{= 10}

⇔p(x−9)2_{+ (y−4)}2 _{= 10−}p_(x−1)2_{+ (y−4)}2

⇔(x₋9)2

+ (y₋4)2

= (x₋1)2

+ (y₋4)2

+ 102

−20p(x₋1)2_{+ (y−4)}2

⇔ −4x₋5 = ₋5p(x₋1)2_{+ (y}

−4)2

⇔16x2

+ 40x+ 25 = 25x2

−50x+ 25 + 25y2

−200y+ 400

⇔9x2

+ 25y2

−90x−200y+ 400 = 0. ✭✺✮

❉✐③❡♠♦s q✉❡ ✭✺✮ é ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❡ss❛ ❡❧✐♣s❡✳ ◆♦t❡ q✉❡✱ s❡ ✉t✐❧✐③❛r♠♦s ❛ té❝♥✐❝❛ ❞❡ ❝♦♠♣❧❡t❛r q✉❛❞r❛❞♦s✱ t❡r❡♠♦s✿

9x2

+ 25y2

−90x₋200y+ 400 = 0

⇔9(x2

−10x+ 25) + 25(y2

−8y+ 16) =−400 + 9·25 + 25·16

⇔9(x−5)2

+ 25(y−4)2

= 225

⇔(x−5)

2

52 +

(y₋4)2

32 = 1. ✭✻✮

❊♠ q✉❡ ✭✻✮ é ❞❡♥♦♠✐♥❛❞❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❡ss❛ ❡❧✐♣s❡✱ ❝✉❥♦ ❝❡♥tr♦ (5,4)✱ ♦ r❛✐♦

♠❛✐♦ra= 5 ❡ ♦ r❛✐♦ ♠❡♥♦r b= 3 ♣♦❞❡♠ s❡r ♦❜s❡r✈❛❞♦s✳ ✷

◆❛ ♣ró①✐♠❛ s❡çã♦ ✈❡r❡♠♦s q✉❡✱ ♣❡❧❛ tr❛♥s❧❛çã♦ ❞❡ ✉♠❛ ❝✉r✈❛ ♣♦r ✉♠ ❞❛❞♦ ✈❡t♦r✱ ♣♦❞❡✲s❡ r❡❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❞❡ ✉♠ ❝ô♥✐❝❛ ❝♦♠ ♦ ❝❡♥tr♦ ❡♠ q✉❛❧q✉❡r ♣♦♥t♦ ❞♦ ♣❧❛♥♦✱ ♦✉ ❛✐♥❞❛✱ q✉❡ ❞❛❞❛ ✉♠❛ ❝ô♥✐❝❛ ❝✉❥♦ ❝❡♥tr♦ ♥ã♦ é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ♣♦❞❡✲s❡ tr❛♥s❧❛❞á✲❧❛ ❞❡ ♠♦❞♦ q✉❡ ✐ss♦ ♦❝♦rr❛✳

❉❡ ♠♦❞♦ ❣❡r❛❧✱ ✈❡r❡♠♦s q✉❡ ♣❡❧❛ tr❛♥s❧❛çã♦ ❞❛ ❡❧✐♣s❡ ❝❡♥tr❛❞❛ ❡♠ (0,0) ♣♦r ✉♠

✈❡t♦r ~v = (x0, y0)✱ ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❡ ✉♠❛ ❡❧✐♣s❡ ❝❡♥tr❛❞❛ ❡♠ (x0, y0)✱ ❝♦♠ r❡t❛

❢♦❝❛❧ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ Ox✱ é (x−x0) 2

a2 +

(y₋y0)2

b2 = 1✳ ◆♦ ❝❛s♦ ❞❛ r❡t❛ ❢♦❝❛❧ s❡r

♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦Oy✱ tr♦❝❛✲s❡ a ♣♦rb✳

❊①❡♠♣❧♦ ✹✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❛ ❡❧✐♣s❡ ❝♦♠ ❝❡♥tr♦C = (2,₋1)✱ r❛✐♦ ♠❛✐♦r

✐❣✉❛❧ ❛4 ❡ r❛✐♦ ♠❡♥♦r ✐❣✉❛❧ ❛ 3✱ s❛❜❡♥❞♦ q✉❡ ❛ r❡t❛ ❢♦❝❛❧ é ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ Oy✳

❙♦❧✉çã♦✿ ❉❛❞♦ q✉❡ ❛ r❡t❛ ❢♦❝❛❧ é ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ Oy✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❡q✉❛✲

çã♦ ❞❡ss❛ ❡❧✐♣s❡ ♥❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ❥á q✉❡ ❛s ♠❡❞✐❞❛s ❞♦ r❛✐♦ ♠❛✐♦r ❡ ❞♦ ♠❡♥♦r sã♦ ❝♦♥❤❡❝✐❞❛s✳ P♦rt❛♥t♦✱(x−2)

2

32 +

(y₋(₋1))2

42 = 1 é ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❡ss❛ ❡❧✐♣s❡✳

❉❡s❡♥✈♦❧✈❡♥❞♦ ❛s ♣♦tê♥❝✐❛s ❡ ❛❣r✉♣❛♥❞♦ ♦s t❡r♠♦s s❡♠❡❧❤❛♥t❡s✱ ❞❡t❡r♠✐♥❛♠♦s s✉❛ ❡q✉❛çã♦ ❣❡r❛❧✿

x2

−4x+ 4 9 +

y2

+ 2y+ 1 16 = 1

⇔16x2−64x+ 64 + 9y2+ 18y+ 9−144 = 0

⇔16x2

+ 9y2

−64x+ 18y₋71 = 0.

▲♦❣♦✱ 16x2

+ 9y2

−64x+ 18y−71 = 0 é ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❡ss❛ ❡❧✐♣s❡✳ ✷

❊①❡♠♣❧♦ ✺✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ ❡❧✐♣s❡ ❝✉❥♦s ❢♦❝♦s sã♦ ♦s ♣♦♥t♦sA= (−2√6,−2√6)

❡B = (2√6,2√6)✱ t❛❧ q✉❡ ❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❞❡ ✉♠ ♣♦♥t♦ ❞❛ ❡❧✐♣s❡ ❛ ❝❛❞❛ ✉♠ ❞♦s

❢♦❝♦s é2a= 16✳

❙♦❧✉çã♦✿ ❆♥t❡s ♠❡s♠♦ ❞❡ ❞❡t❡r♠✐♥❛r♠♦s ❛ ❡q✉❛çã♦✱ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r ❛❧❣✉♥s ❞♦s ❡❧❡♠❡♥t♦s ❞❡ss❛ ❡❧✐♣s❡✳ ❈♦♠♦ sã♦ ❝♦♥❤❡❝✐❞♦s ♦s ❢♦❝♦s ❞❡ss❛ ❡❧✐♣s❡✱ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡❧❡s✿

d(A, B) =

q

(2√6 + 2√6)2_{+ (2}√_{6 + 2}√₆₎2 ₌√_{192 = 8}√_{3 = 2c.}

❈♦♠♦ a = 8✱ c = 4√3 ❡ b2

= c2

−a2✱ s❡❣✉❡ q✉❡

b = 4 ❡ e =

√

3

2 ✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦

❝♦♥❤❡❝❡♠♦s ♦s ❢♦❝♦s✱ ♣♦❞❡♠♦s ❢❛❝✐❧♠❡♥t❡ ❞❡t❡r♠✐♥❛r ♦ ❝❡♥tr♦ ❞❛ ❡❧✐♣s❡ ❡ ❛ r❡t❛ ❢♦❝❛❧✳ ❊ss❛ ❡❧✐♣s❡ t❡♠ ❝❡♥tr♦(0,0)❡ r❡t❛ ❢♦❝❛❧ r:x−y= 0✳

x

O

A

B

y

=

_x

❋✐❣✉r❛ ✻✿ ❊❧✐♣s❡ ❝♦♠ ❢♦❝♦sA = (₋2√6,₋2√6) ❡B = (2√6,2√6)✳

P❡❧❛ ❞❡✜♥✐çã♦✱ d(P, A) +d(P, B) = 2a✱ ❛ss✐♠✿ q

(x+ 2√6)2_{+ (y+ 2}√₆₎2₊ q

(x−2√6)2_{+ (y−2}√₆₎2 _{= 16}

⇔(x+ 2√6)2

+ (y+ 2√6)2

= 162

+ (x−2√6)2

+ (y−2√6)2

−32

q

(x−2√6)2_{+ (y−2}√₆₎2

⇔4√6x+ 4√6y= 256−4√6x−4√6y−32

q

(x−2√6)2_{+ (y−2}√₆₎2

⇔8√6x+ 8√6y₋256 =₋32

q

(x₋2√6)2_{+ (y−2}√₆₎2

⇔√6x+√6y₋32 =₋4

q

(x₋2√6)2_{+ (y}

−2√6)2

⇔6x2

+ 6y2

+ 1024₋64√6x+ 12xy₋64√6y= 16(x2

−4√6x+y2

−4√6y+ 48)

⇔10x2

−12xy+ 10y2

−256 = 0

⇔5x2

−6xy+ 5y2

−128 = 0. ✭✼✮

▲♦❣♦✱ ✭✼✮ é ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❡ss❛ ❡❧✐♣s❡✳ ✷

❖❜s❡r✈❡ q✉❡✱ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ss❛ ❡❧✐♣s❡✱ ❛♣❛r❡❝❡✉ ✉♠ t❡r♠♦ q✉❛✲ ❞rát✐❝♦ ♠✐st♦✱ ✉♠ t❡r♠♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❢♦r♠❛❞♦ ♣♦r ❞✉❛s ✈❛r✐á✈❡✐s✱ ♦ t❡r♠♦ 6xy✱

q✉❡ ❛✐♥❞❛ ♥ã♦ ❤❛✈✐❛ ❛♣❛r❡❝✐❞♦ ❡♠ ♥♦ss♦s ❡①❡♠♣❧♦s✳ P❡r❝❡❜❛ q✉❡ ❡st❡ é ♦ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦ ❡♠ q✉❡ ❛ ❝ô♥✐❝❛ ❡st✉❞❛❞❛ ♥ã♦ t❡♠ s✉❛ r❡t❛ ❢♦❝❛❧ ♣❛r❛❧❡❧❛ ❛ ♥❡♥❤✉♠ ❞♦s ❡✐①♦s ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛❞♦t❛❞♦✳

❈♦♠♦ ❛ ❝ô♥✐❝❛ ❢♦✐ ❝♦♥str✉í❞❛ ❝♦♥❤❡❝❡♥❞♦✲s❡ ♦s ❢♦❝♦s ❡ ❛ ❝♦♥st❛♥t❡ 2a✱ é ♣♦ssí✈❡❧

✐❞❡♥t✐✜❝❛r t♦❞♦s ♦s s❡✉s ❡❧❡♠❡♥t♦s✳ ❊♥tr❡t❛♥t♦✱ s❡ ♥♦s ❢♦ss❡ ❞❛❞❛ ❛ ❡q✉❛çã♦ ❝♦♠ ♦

♦❜❥❡t✐✈♦ ❞❡✱ ❛ ♣❛rt✐r ❞❡❧❛✱ ✐❞❡♥t✐✜❝❛r♠♦s s❡✉s ❡❧❡♠❡♥t♦s✱ ♦ ♣r♦❜❧❡♠❛ s❡ t♦r♥❛r✐❛ ♠❛✐s ❡❧❛❜♦r❛❞♦✱ ❝♦♠♦ ✈❡r❡♠♦s ❛té ♦ ✜♥❛❧ ❞❡ss❡ tr❛❜❛❧❤♦✳

✶✳✸ ❍✐♣ér❜♦❧❡

❆❣♦r❛ ❢❛❧❛r❡♠♦s s♦❜r❡ ❛ ❤✐♣ér❜♦❧❡✱ s✉❛ ❞❡✜♥✐çã♦ ❡ ❛❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s✳

❉❡✜♥✐çã♦ ✸✳ ❍✐♣ér❜♦❧❡ é ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ❞❡ ✉♠ ♣❧❛♥♦ ❝✉❥♦ ✈❛❧♦r ❛❜s♦✲ ❧✉t♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❞❛s ❞✐stâ♥❝✐❛s ❛ ❞♦✐s ♣♦♥t♦s ✜①♦s ✭❢♦❝♦s✮ é ❝♦♥st❛♥t❡ ❡ ♠❡♥♦r ❞♦ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❢♦❝♦s✳

❖✉ s❡❥❛✱ s❡ P é ✉♠ ♣♦♥t♦ ❞❛ ❤✐♣ér❜♦❧❡ q✉❡ t❡♠ ♣♦r ❢♦❝♦s ♦s ♣♦♥t♦s A ❡ B✱ ❡♥tã♦

|d(P, A)₋d(P, B)_| = 2a✱ ❡♠ q✉❡ a é ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧ ♣♦s✐t✐✈❛ t❛❧ q✉❡ a < c✱ C= (a, b)✱ D= (0, b)✱ E = (a,0)❡ OC =c

x y

(-c,0)=A B=(c,0)

D C

E O

❋✐❣✉r❛ ✼✿ ❍✐♣ér❜♦❧❡ ❝ê♥tr✐❝❛✳

❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ❝♦♥s✐❞❡r❡♠♦s ✉♠❛ ❤✐♣ér❜♦❧❡ ❝❡♥tr❛❞❛ ♥❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ❞❡ t❛❧ ♠♦❞♦ q✉❡ ♦s ❢♦❝♦s ❡stã♦ ❧♦❝❛❧✐③❛❞♦s s♦❜r❡ ♦ ❡✐①♦ Ox✳

❙❡❥❛♠ A = (−c,0) ❡ B = (c,0) ♦s ❢♦❝♦s ❞❡ss❛ ❡❧✐♣s❡✳ ❆ss✐♠✱ ♣❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ❛

❡q✉❛çã♦ r❡❞✉③✐❞❛ ♣❛r❛ ♦ ❝❛s♦ ❞❛ ❞✐❢❡r❡♥ç❛ s❡r ♣♦s✐t✐✈❛✱ ✷❛✱ ❢❛③❡♠♦s✿

⇔p(x+c)2_+y2 ₌p_(x−c)2 _+y2_{+ 2a}

⇔(x+c)2

+y2

= (x₋c)2

+y2

+ 4a2

+ 4ap(x₋c)2_+y2

⇔cx₋a2

=ap(x₋c)2_+y2

⇔x2

(c2

−a2

)₋y2 a2

=a2 b2

.

❈♦♠♦c2

−a2

=b2✱ t❡♠♦s q✉❡ x2

b2

−y2 a2

=a2

b2✳ ❉✐✈✐❞✐♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛

✐❣✉❛❧❞❛❞❡ ♣♦ra2

b2_{✱ t❡r♠♦s✿}

x2 a2 −

y2

b2 = 1. ✭✽✮

▲♦❣♦✱ ✭✽✮ é ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ♣❛r❛ ❛ ❤✐♣ér❜♦❧❡ ❝✉❥♦ ❝❡♥tr♦ é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡ ❛ r❡t❛ ❢♦❝❛❧ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ❡✐①♦ Ox✳ ❆q✉✐ ❢♦✐ ❢❡✐t❛ ❛ ❛♥á❧✐s❡ ♣❛r❛ ♦

r❛♠♦ ❞✐r❡✐t♦ ❞❛ ❤✐♣ér❜♦❧❡✱ ♣❛r❛ ♦ ❝❛s♦ ❞❛ ❞✐❢❡r❡♥ç❛ s❡r ♣♦s✐t✐✈❛✳ ❊♥tr❡t❛♥t♦✱ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❡♥❝♦♥tr❛♠♦s ❛ ♠❡s♠❛ ❡q✉❛çã♦ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ♦ ❝❛s♦ ❞❛ ❞✐❢❡r❡♥ç❛ s❡r ♥❡❣❛t✐✈❛✱ ✐st♦ é✱ ♣❛r❛ ♦ r❛♠♦ ❡sq✉❡r❞♦ ❞❛ ❤✐♣ér❜♦❧❡✳

❙❡❣✉✐♥❞♦ t❛♠❜é♠ ♦ ♠❡s♠♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ q✉❛♥❞♦ ❡s❝♦❧❤✐❞❛ ✉♠❛ ❤✐♣ér❜♦❧❡ ❝❡♥✲ tr❛❞❛ ❡♠(0,0)t❛❧ q✉❡ ❛ r❡t❛ ❢♦❝❛❧ ❝♦✐♥❝✐❞❛ ❝♦♠ ♦ ❡✐①♦Oy✱ ❡♥❝♦♥tr❛♠♦s ❝♦♠♦ ❡q✉❛çã♦

❝❛♥ô♥✐❝❛ ❞❛ ❤✐♣ér❜♦❧❡✱ ❛ ❡q✉❛çã♦

y2 a2 −

x2

b2 = 1. ✭✾✮

❊①❡♠♣❧♦ ✻✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❛ ❤✐♣ér❜♦❧❡ ❝✉❥♦s ❢♦❝♦s sã♦ A = (0,2√5) ❡

B = (0,₋2√5)✱ t❛❧ q✉❡ ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❞❛s ❞✐stâ♥❝✐❛s ❞❡ ✉♠ ♣♦♥t♦ ❞❛

❤✐♣ér❜♦❧❡ ❛ ❝❛❞❛ ✉♠ ❞♦s ❢♦❝♦s é2a= 8✳

❙♦❧✉çã♦✿ ❈♦♠♦ sã♦ ❝♦♥❤❡❝✐❞♦s ♦s ❢♦❝♦s ❞❛ ❤✐♣ér❜♦❧❡✱ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ❛ ❞✐stâ♥✲ ❝✐❛ ❡♥tr❡ ❡❧❡s✱ ♦ ❝❡♥tr♦ ❞❛ ❤✐♣ér❜♦❧❡✱ ❜❡♠ ❝♦♠♦ ❛ r❡t❛ ❢♦❝❛❧✳ ◆♦ ❝❛s♦✱ s❡d(A, B) = 2c✱

❡♥tã♦ c = 2√5✳ ❆❧é♠ ❞✐ss♦✱ ♦ ❝❡♥tr♦ é ♦ ♣♦♥t♦ (0,0) ❡ t❡♠♦s q✉❡ x = 0 é ❛ r❡t❛ q✉❡

♣❛ss❛ ♣❡❧♦s ❢♦❝♦s✳ ❈♦♠♦ a = 4✱ c= 2√5 ❡ b2

= c2

−a2_{✱ s❡❣✉❡ q✉❡}

b = 2✳ ❈♦❧♦❝❛♥❞♦

❡ss❛s ✐♥❢♦r♠❛çõ❡s ♥❛ ❡q✉❛çã♦ ✭✾✮✱ t❡♠♦s ❝♦♠♦ ❡q✉❛çã♦ ❞❡ss❛ ❤✐♣ér❜♦❧❡✱ ❛ ❡q✉❛çã♦

y2

42 − x2

22 = 1.

❉❡s❡♥✈♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈ê✲❧❛ ❝♦♠♦ 4x2

−y2

+ 16 = 0✱ ❡♠ q✉❡

❡ss❛ é ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❡ss❛ ❤✐♣ér❜♦❧❡✳ ✷

❊①❡♠♣❧♦ ✼✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❛ ❤✐♣ér❜♦❧❡ ❡♠ q✉❡A= (2,5)❡B = (₋2,3)

sã♦ ♦s ❢♦❝♦s ❡ ♦ ♠ó❞✉❧♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ❞✐stâ♥❝✐❛s ❞❡ ✉♠ ♣♦♥t♦P ❞❛ ❤✐♣ér❜♦❧❡

❛ ❝❛❞❛ ✉♠ ❞♦s ❢♦❝♦s é2a=√5✳

❙♦❧✉çã♦✿ ❈♦♥❤❡❝✐❞♦s ♦s ❢♦❝♦s ❡ ♦ ♠ó❞✉❧♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ❞✐stâ♥❝✐❛s✱ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r ❛❧❣✉♥s ❞♦s ♣r✐♥❝✐♣❛✐s ❡❧❡♠❡♥t♦s ❞❡ss❛ ❤✐♣ér❜♦❧❡✱ ❝♦♠♦ ❛ r❡t❛ ❢♦❝❛❧✱ ❛ r❡t❛ ♥ã♦ ❢♦❝❛❧ ❡ ❛ ❞✐stâ♥❝✐❛ ❢♦❝❛❧ 2c✳ ❆❧é♠ ❞✐ss♦✱ s❡ ✉t✐❧✐③❛r♠♦s ♦ ❢❛t♦ ❞❡ q✉❡ b2

=c2

−a2✱

♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r b✱ ♣♦r ❥á s❡r❡♠ ❝♦♥❤❡❝✐❞♦s ♦s ✈❛❧♦r❡s ❞❡ a ❡ c✳ P❡❧❛ ❞❡✜♥✐çã♦✱ d(P, A)−d(P, B) = 2a✱ s❡❣✉❡ q✉❡✿

p

(x₋2)2_{+ (y−5)}2₋p_{(x+ 2)}2_{+ (y−3)}2 ₌√₅

⇔p(x₋2)2_{+ (y}

−5)2 ₌√_{5 +}p_{(x+ 2)}2_{+ (y}

−3)2

⇔(x₋2)2

+ (y₋5)2

= 5 + (x+ 2)2

+ (y₋3)2

+ 2√5p(x+ 2)2_{+ (y}

−3)2

⇔ −8x−4y+ 11 = 2√5p(x+ 2)2_{+ (y−3)}2

⇔64x2

+ 16y2

+ 121 + 64xy−176x−88y= 20(x2

+ 4x+ 4 +y2

−6y+ 9)

⇔44x2

+ 64xy₋4y2

−256x+ 32y₋139 = 0.

O x

y

B

A

❋✐❣✉r❛ ✽✿ ❍✐♣ér❜♦❧❡ ❝♦♠ ❢♦❝♦sA= (2,5) ❡B = (−2,3)✳

❉❡t❡r♠✐♥❛❞❛ ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❛ ❤✐♣ér❜♦❧❡✱ ♥♦t❛✲s❡ ♠❛✐s ✉♠❛ ✈❡③ ❛ ♦❝♦rrê♥❝✐❛ ❞♦ t❡r♠♦ q✉❛❞rát✐❝♦ ♠✐st♦✱ 64xy✳ ◆❛ s❡çã♦ ✷ ❞❡ss❡ tr❛❜❛❧❤♦ ✈❡r❡♠♦s ❝♦♠♦✱ ♣♦r ♠❡✐♦ ❞❡

✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ❡✐①♦s✱ ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❡ss❛ ❝ô♥✐❝❛ ❞❡ ♠♦❞♦ q✉❡ s❡❥❛ ♣♦ssí✈❡❧ ✐❞❡♥t✐✜❝❛r s❡✉s ❡❧❡♠❡♥t♦s✳✷

✶✳✹ P❛rá❜♦❧❛

◆❡ss❛ s❡çã♦ ❡st✉❞❛r❡♠♦s ❛ ♣❛rá❜♦❧❛ ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳

❉❡✜♥✐çã♦ ✹✳ P❛rá❜♦❧❛ é ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ❞❡ ✉♠ ♣❧❛♥♦ q✉❡ ❡q✉✐❞✐st❛♠ ❞❡ ✉♠❛ r❡t❛ ❡ ✉♠ ♣♦♥t♦✱ ❢♦r❛ ❞❡❧❛✱ ❞❛❞♦s✳ ❆ r❡t❛ é ❞❡♥♦♠✐♥❛❞❛ ❣❡r❛tr✐③ ❞❛ ♣❛rá❜♦❧❛✱ ❡♥q✉❛♥t♦ ♦ ♣♦♥t♦ é s❡✉ ❢♦❝♦✳

❖✉ s❡❥❛✱ s❡ P é ✉♠ ♣♦♥t♦ ❞❛ ♣❛rá❜♦❧❛ ❝✉❥❛ r❡t❛ ❞✐r❡tr✐③ é r ❡ ♦ ❢♦❝♦ é ♦ ♣♦♥t♦ A✱ ♣♦r ❞❡✜♥✐çã♦ d(P, r) = d(P, A)✳ P❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❡ ✉♠❛

♣❛rá❜♦❧❛✱ ♣r❡❝✐s❛♠♦s r❡❝♦r❞❛r ❝♦♠♦ ❞❡t❡r♠✐♥❛r ❛ ❞✐stâ♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ ❛ ✉♠❛ r❡t❛✳

Pr♦♣♦s✐çã♦ ✶✳ ❉❛❞♦s ✉♠ ♣♦♥t♦ P = (x0, y0) ❡ ✉♠❛ r❡t❛ r :ax+by−c= 0✱ ❡♠ ✉♠

♣❧❛♥♦✱ t❡♠✲s❡ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❞❡P ❛ r é ❞❛❞❛ ♣♦r d(P, r) = |ax0√+by0−c|

a2 _+b2 ✳

❆ ❞❡♠♦♥str❛çã♦ ♣❛r❛ ❡ss❛ ♣r♦♣♦s✐çã♦ ♣♦❞❡ s❡r ❢❡✐t❛ ✉t✐❧✐③❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá✲ ❣♦r❛s✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ❢❛t♦ ❞❡ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ ❛ ✉♠❛ r❡t❛ é ❛ ♠❡♥♦r ❞❛s ❞✐stâ♥❝✐❛s✳ ❆ss✐♠✱ t♦♠❛♥❞♦ ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❛ r❡t❛ ❡ ♦ ♣é ❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❜❛✐①❛❞❛ ❞♦ ♣♦♥t♦ ❛té ❛ r❡t❛✱ é ♣♦ssí✈❡❧ ❝♦♥❝❧✉✐r ❡ss❛ ❞❡♠♦♥str❛çã♦✳ ▼❛s✱ ♥ã♦ é ❛ ✐♥t❡♥çã♦ ♥♦ ♠♦♠❡♥t♦ ❢❛③ê✲❧❛✱ ❡ s✐♠ ❡st✉❞❛r♠♦s ❛ ♣❛rá❜♦❧❛✳

❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ❝♦♥s✐❞❡r❡♠♦s ✉♠❛ ♣❛rá❜♦❧❛ ❝✉❥♦ ❢♦❝♦A=0,c b

❡stá s♦❜r❡ ♦ ❡✐①♦ Oy ❡ q✉❡ t❡♠ ❛ r❡t❛ r :y = −c

b ♣♦r s✉❛ ❞✐r❡tr✐③✳ ◆♦t❡ q✉❡ ❡ss❡s ✈❛❧♦r❡s

❢♦r❛♠ ❡s❝♦❧❤✐❞♦s ❞❡ ♠♦❞♦ q✉❡ ♦ ✈ért✐❝❡ ❞❛ ♣❛rá❜♦❧❛ ❝♦✐♥❝✐❞❛ ❝♦♠ ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛❞♦t❛❞♦✱ ✐st♦ é✱ s❡ V é ♦ ✈ért✐❝❡ ❞❡ss❛ ♣❛rá❜♦❧❛✱ ❡♥tã♦V = (0,0)✳

x

y

O

P

r

:

by

+

c

=0

A

❋✐❣✉r❛ ✾✿ P❛rá❜♦❧❛ ❝♦♠ ✈ért✐❝❡ (0,0)✳

❙❡♥❞♦ P = (x, y)✉♠ ♣♦♥t♦ ❞❡ss❛ ♣❛rá❜♦❧❛✱ ♣♦r ❞❡✜♥✐çã♦ d(P, r) =d(P, A)✱ ❡♥tã♦

|0x+by+c| √

02_+b2 = r

(x₋0)2_+y

− c_b

2

._⇔ (by+c) 2

b2 =x

2

+y₋ c b

2

⇔b2 y2

+c2

+ 2bcy=b2 x2

+b2 y2

+c2

−2bcy

⇔4bcy=b2 x2

⇔y= x

2

4c/b

P❡r❝❡❜❛ q✉❡ 2c

b é ❛ ❞✐stâ♥❝✐❛ ❞♦ ❢♦❝♦ à r❡t❛ ❞✐r❡tr✐③✱ ❛ q✉❛❧ ❞❡♥♦♠✐♥❛r❡♠♦s ♣❛râ✲

♠❡tr♦ ❞❛ ♣❛rá❜♦❧❛ ❡ r❡♣r❡s❡♥t❛r❡♠♦s ♣♦rp✳ P♦rt❛♥t♦ ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❛ ♣❛rá❜♦❧❛

q✉❡ t❡♠ ❛ r❡t❛ ❞✐r❡tr✐③ ♥❛ ♠❡s♠❛ ❞✐r❡çã♦ ❞♦ ❡✐①♦Ox ❡ ❝✉❥♦ ✈ért✐❝❡ é ♦ ♣♦♥t♦(0,0) é

y= x

2

2p✳

❙❡ ❛ r❡t❛ ❞✐r❡tr✐③ t✐✈❡r ❛ ♠❡s♠❛ ❞✐r❡çã♦ ❞♦ ❡✐①♦ Oy ❡ ✈ért✐✈❡ (0,0)✱ ❡♥tã♦ ❛ s✉❛

❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ s❡ráx= y

2

2p✱ ❡♠ q✉❡p❝♦rr❡s♣♦♥❞❡ à ❞✐stâ♥❝✐❛ ❞♦ ❢♦❝♦ à r❡t❛ ❞✐r❡tr✐③✳

❊①❡♠♣❧♦ ✽✳ ❉❡t❡r♠✐♥❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦✱ ❛ ❡q✉❛çã♦ ❞❛ ♣❛rá❜♦❧❛ ❡♠ q✉❡ x = ₋3 é ❛

r❡t❛ ❞✐r❡tr✐③ ❡ (3,0) ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ❢♦❝♦ ❞❡ss❛ ♣❛rá❜♦❧❛✳

❙♦❧✉çã♦✿ P❡❧❛ ❞❡✜♥✐çã♦✱ ❛ ♣❛rá❜♦❧❛ é ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s P = (x, y) t❛❧ q✉❡

d(P, A) = d(P, r)✱ ❡♠ q✉❡A❡rsã♦ ♦ ❢♦❝♦ ❡ ❛ r❡t❛ ❞✐r❡tr✐③ ❞❛ ♣❛rá❜♦❧❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❆ss✐♠✱

⇔p(x−3)2_{+ (y−0)}2 ₌ |1x_√+ 0y−3|

12_{+ 0}2

⇔px2

−6x+ 9 +y2 ₌

|1x₋3_|

⇔x2

−6x+ 9 +y2

=x2

−6x+ 9

⇔x= y

2

12.

P♦rt❛♥t♦✱ x= y

2

2_·6 é ❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❡ss❛ ♣❛rá❜♦❧❛✳✷

❱❡r❡♠♦s ♥❛ s❡çã♦ s❡❣✉✐♥t❡ q✉❡✱ tr❛♥s❧❛❞❛♥❞♦ ❛ ♣❛rá❜♦❧❛ ♣♦r ✉♠ ❞❛❞♦ ✈❡t♦r✱ ♣♦❞❡✲ ♠♦s ❡s❝r❡✈❡r s✉❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛✱ ❝✉❥♦ ✈ért✐❝❡(x0, y0)é ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞♦ ♣❧❛♥♦✱

♣❡❧❛s s❡❣✉✐♥t❡s ❡q✉❛çõ❡s✿

y₋y0 = (x−x0)

2

2p ❡ x−x0 =

(y₋y0)2

2p ✭✶✵✮

❊♠ q✉❡ ❛ ♣r✐♠❡✐r❛ é ♣❛r❛ ♦ ❝❛s♦ ❞❛ r❡t❛ ❞✐r❡tr✐③ t❡r ♠❡s♠❛ ❞✐r❡çã♦ ❞♦ ❡✐①♦ Ox✱ ❡

❛ s❡❣✉♥❞❛ ♣❛r❛ ♦ ❝❛s♦ ❞❛ r❡t❛ ❞✐r❡tr✐③ t❡r ♠❡s♠❛ ❞✐r❡çã♦ ❞♦ ❡✐①♦ Oy✳

❊①❡♠♣❧♦ ✾✳ ❉❛❞❛ ❛ ♣❛rá❜♦❧❛ ϕ ❞❡ ❡q✉❛çã♦ x2

−8x₋2y+ 20 = 0✱ ❞❡t❡r♠✐♥❡ s✉❛

❡q✉❛çã♦ ❝❛♥ô♥✐❝❛✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈ért✐❝❡ ❡ ♦ ♣❛râ♠❡tr♦p ✭❞✐stâ♥❝✐❛ ❞♦ ❢♦❝♦ à r❡t❛

❞✐r❡tr✐③✮✳

❙♦❧✉çã♦✿ ❘❡❡s❝r❡✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❡ ❝♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦s✱ t❡♠♦s✿

x2

−8x−2y+ 20 = 0

⇔2y−20 =x2

−8x

⇔2y₋20 + 16 = x2

−8x+ 16

⇔2(y₋2) = (x₋4)2

⇔(y−2) = (x−4)

2

2 .

❈♦♠♣❛r❛♥❞♦ ❝♦♠ ❛ ❡q✉❛çã♦ ✭✶✵✮✱ t❡♠♦s q✉❡(4,2)é ♦ ✈ért✐❝❡ ❞❡ss❛ ♣❛rá❜♦❧❛ ❡p= 1

s❡✉ ♣❛râ♠❡tr♦✳ ✷

❊①❡♠♣❧♦ ✶✵✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❛ ♣❛rá❜♦❧❛ ❝♦♠ ❢♦❝♦ F = (2,2) ❡ r❡t❛

❞✐r❡tr✐③r:x+ 2y= 1✳

❙♦❧✉çã♦✿ P❡❧❛ ❞❡✜♥✐çã♦✱ ✉♠❛ ♣❛rá❜♦❧❛ é ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ❡q✉✐❞✐st❛♥t❡s ❛♦ ❢♦❝♦ ❡ à r❡t❛ ❞✐r❡tr✐③✱ ♣♦rt❛♥t♦✱ t❡♠♦sd(P, F) =d(P, r) ♣❛r❛ t♦❞♦ ♣♦♥t♦P = (x, y)

❞❛ ♣❛rá❜♦❧❛✳ ▲♦❣♦✱

d(P, F) =d(P, r)

⇔p(x₋2)2_{+ (y}

−2)2 ₌ |1x_√+ 2y−1|

12_{+ 2}2

⇔x2

−4x+ 4 +y2

−4y+ 4 = (x+ 2y−1)

2

5

⇔5x2

+ 5y2

−20x₋20y+ 40 =x2

+ 4y2

+ 1 + 4xy₋2x₋4y

⇔4x2

−4xy+y2

−18x₋16y+ 39 = 0. ✭✶✶✮

r

x O

y

F

❋✐❣✉r❛ ✶✵✿ P❛rá❜♦❧❛ ❝♦♠ r❡t❛ ❞✐r❡tr✐③ r:x+ 2y = 1✳

P♦rt❛♥t♦✱ ✭✶✶✮ é ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❛ ♣❛rá❜♦❧❛ ϕ✳ ◆♦t❡ ♠❛✐s ✉♠❛ ✈❡③ ❛ ♦❝♦rrê♥❝✐❛

❞♦ t❡r♠♦ q✉❛❞rát✐❝♦ ♠✐st♦✳✷

✷ ❱❡t♦r❡s ♥♦ ♣❧❛♥♦ ❡ tr❛♥s❧❛çã♦

❉❡ ♦r✐❣❡♠ ♥♦ ❧❛t✐♠ ✈❡❝t♦r ♦✉ ✈❡❤❡r❡✱ ✈❡t♦r é ♦ q✉❡ tr❛♥s♣♦rt❛✱ ♦ q✉❡ ❧❡✈❛✳ ◆❡ss❛ ♣❛rt❡ ❞♦ tr❛❜❛❧❤♦ s❡rá ❛❜♦r❞❛❞❛ s✉❛ ❛♣❧✐❝❛çã♦ ♥❛ tr❛♥s❧❛çã♦ ❞❡ ♣♦♥t♦s ❡♠ ✉♠ ♣❧❛♥♦✳ ❊♥tr❡t❛♥t♦✱ ♦ q✉❡ é ✈❡t♦r❄ P❛r❛ ❡♥t❡♥❞❡r♠♦s ♦ s❡✉ ❝♦♥❝❡✐t♦✱ ❞❡✈❡♠♦s r❡t♦♠❛r ❛s ❞❡✜♥✐çõ❡s ❞❡ s❡❣♠❡♥t♦s ❞❡ r❡t❛ ♦r✐❡♥t❛❞♦s ❡q✉✐♣♦❧❡♥t❡s✳

❈♦♥s✐❞❡r❡✱ ❡♠ ✉♠ ♣❧❛♥♦✱ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡♠ q✉❡ ♦s ♣♦♥t♦s A ❡ B sã♦

t❛✐s q✉❡ A = (a1, a2) ❡ B = (b1, b2)✳ ❙❡❣✉♥❞♦ ❊❧♦♥ ❡♠ ❬✺❪✱ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ é

♦r✐❡♥t❛❞♦ q✉❛♥❞♦ s❡ ❡st❛❜❡❧❡❝❡ q✉❛❧ ❞❡ s✉❛s ❡①tr❡♠✐❞❛❞❡s é ♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❡ q✉❛❧ é ♦ ♣♦♥t♦ ✜♥❛❧✳ P♦r ❡①❡♠♣❧♦✱ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ♦r✐❡♥t❛❞♦ AB é ❛q✉❡❧❡ ❡♠ q✉❡ A é ♦

♣♦♥t♦ ✐♥✐❝✐❛❧ ❡ B✱ ♦ ✜♥❛❧✳ ❊♠ ❝♦♥tr❛ ♣❛rt✐❞❛✱ ✜❝❛ s✉❜t❡♥❞✐❞♦ q✉❡ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛

♦r✐❡♥t❛❞♦BAé ❛q✉❡❧❡ ❡♠ q✉❡Bé ♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❡A✱ ♦ ✜♥❛❧✳ P❛r❛ s✐♠♣❧✐✜❝❛r ❛ ❡s❝r✐t❛✱

✉t✐❧✐③❛r❡♠♦s ❛♣❡♥❛s ✏s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦✑ ♥♦ ❧✉❣❛r ❞❡ ✏s❡❣♠❡♥t♦ ❞❡ r❡t❛ ♦r✐❡♥t❛❞♦✑✳ ❙♦❜r❡ ♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦✱ t❡♠✲s❡ q✉❡ s❡✉ ❝♦♠♣r✐♠❡♥t♦ ♣♦❞❡ s❡r ♦❜t✐❞♦ q✉❛♥❞♦ sã♦ ❝♦♥❤❡❝✐❞❛s ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ s✉❛s ❡①tr❡♠✐❞❛❞❡s✳ P♦r ❞❡✜♥✐çã♦✱ ❡ss❡ ❝♦♠♣r✐♠❡♥t♦ ❝♦rr❡s♣♦♥❞❡ à ❞✐stâ♥❝✐❛ ❡♥tr❡ s✉❛s ❡①tr❡♠✐❞❛❞❡s✳ ❆ss✐♠✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ AB é ✐❣✉❛❧ ❛ p(b1−a1)2_{+ (b2−a2)}2✱ ❝♦♠♦ ✈✐st♦ ♥❛ s❡çã♦ ✶✳✶✳ ◆♦t❡ q✉❡

✉♠ s❡❣♠❡♥t♦ ❞❡❣❡♥❡r❛❞♦✱ q✉❡ t❡♥❤❛ ❝♦♠♦ ♣♦♥t♦s ✐♥✐❝✐❛❧ ❡ ✜♥❛❧ ♦ ♠❡s♠♦ ♣♦♥t♦✱ é ❞❡♥♦♠✐♥❛❞♦ ♥✉❧♦ ❡ s❡✉ ❝♦♠♣r✐♠❡♥t♦ é ③❡r♦✳

◗✉❛♥t♦ à ❞✐r❡çã♦ ❞❡ ✉♠ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦✱ ❡ss❛ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛ à ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ s✉♣♦rt❡ ❞♦ s❡❣♠❡♥t♦✳ P♦rt❛♥t♦✱ ❞♦✐s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s tê♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦ s❡ s✉❛s r❡t❛s s✉♣♦rt❡s t✐✈❡r❛♠ ❛ ♠❡s♠❛ ✐♥❝❧✐♥❛çã♦✱ ✐st♦ é✱ ♦✉ ❡ss❡s s❡❣♠❡♥t♦s sã♦ ❝♦❧✐♥❡❛r❡s ♦✉ ❡stã♦ ❝♦♥t✐❞♦s ❡♠ r❡t❛s ♣❛r❛❧❡❧❛s✳

❙❡AB ❡CD ❢♦r❡♠ ❝♦❧✐♥❡❛r❡s ❡ ✐♥❞✉③✐r❡♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦ ❞❛ r❡t❛ q✉❡ ♦s ❝♦♥té♠✱

❡ss❡s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s tê♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦✳ ❈❛s♦ ❝♦♥trár✐♦ ♦s s❡❣♠❡♥t♦s s❡rã♦ ❝♦♥s✐❞❡r❛❞♦s ❞❡ s❡♥t✐❞♦ ❝♦♥trár✐♦✳

A B

C D

(a) mesmo sentido. (b) sentidos contrários.

A B

C

D

❋✐❣✉r❛ ✶✶✿ ❙❡❣♠❡♥t♦s ❝♦❧✐♥❡❛r❡s✳

◆♦ ❝❛s♦ ❞❡ AB ❡ CD ❡st❛r❡♠ ❝♦♥t✐❞♦s ❡♠ r❡t❛s ♣❛r❛❧❡❧❛s✱ s❡ ♦s s❡❣♠❡♥t♦s AC ❡ BD t✐✈❡r❡♠ ✐♥t❡rs❡çã♦ ✈❛③✐❛✱ ❡♥tã♦ AB ❡ CD s❡rã♦ ❞❡ ♠❡s♠❛ ❞✐r❡çã♦❀ ❝❛s♦ ❝♦♥trár✐♦

♦s s❡❣♠❡♥t♦s s❡rã♦ ❝♦♥s✐❞❡r❛❞♦s ❞❡ s❡♥t✐❞♦ ❝♦♥trár✐♦✳

A B

C D

A

B C

D

(a) mesmo sentido. (b) sentidos contrários.

❋✐❣✉r❛ ✶✷✿ ❙❡❣♠❡♥t♦s ♣❛r❛❧❡❧♦s✳

❉❡✜♥✐çã♦ ✺✳ ◗✉❛♥❞♦ ❞♦✐s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s✱AB ❡CD✱ sã♦ ❛♠❜♦s ♥✉❧♦s ♦✉ tê♠ ♦

♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦✱ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦✱ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❡q✉✐♣♦❧❡♥t❡s ❡ r❡♣r❡s❡♥t❛❞♦s ♣♦r AB ∼ CD✱ ❧ê✲s❡ ♦ s❡❣♠❡♥t♦ ❆❇ é ❡q✉✐♣♦❧❡♥t❡ ❛♦ s❡❣♠❡♥t♦ ❈❉✳ ❙❡♥❞♦ ❛ r❡❧❛çã♦

❞❡ ❡q✉✐♣♦❧ê♥❝✐❛ ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✭✐✮ ❘❡✢❡①✐✈❛✿ AB _∼AB❀

✭✐✐✮ ❙✐♠étr✐❝❛✿ AB ∼CD s❡✱ ❡ s♦♠❡♥t❡ s❡✱CD ∼AB❀

✭✐✐✐✮ ❚r❛♥s✐t✐✈❛✿ ❙❡ AB _∼CD ❡CD _∼EF✱ ❡♥tã♦ AB _∼EF✳

Pr♦♣♦s✐çã♦ ✷✳ ❉❛❞♦s ✉♠ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦AB✱ ♥ã♦ ♥✉❧♦✱ ❡ ✉♠ ♣♦♥t♦C✱ ❡①✐st❡ ✉♠

ú♥✐❝♦ ♣♦♥t♦D t❛❧ q✉❡ CD é ❡q✉✐♣♦❧❡♥t❡ ❛AB✳

❉❡♠♦♥str❛çã♦✳ ❈♦♥s❡q✉ê♥❝✐❛ ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ P❧❛♥❛✱ ❡ss❛ ♣r♦♣♦s✐çã♦ é ❞❡✲ ♠♦♥str❛❞❛ ❛♥❛❧✐s❛♥❞♦ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✱ ✈✐st♦ q✉❡ ♣❛r❛ s❡r❡♠ ❡q✉✐♣♦❧❡♥t❡s ❤á ❛ ♥❡✲ ❝❡ss✐❞❛❞❡ ❞❡ t❡r❡♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦✿ ✭✐✮ ♦s s❡❣♠❡♥t♦s sã♦ ❝♦❧✐♥❡❛r❡s❀ ✭✐✐✮ ♦s s❡❣♠❡♥t♦ sã♦ ♣❛r❛❧❡❧♦s✳

✭✐✮ ◆♦ ❝❛s♦ ❞❡ A✱ B ❡ C s❡r❡♠ ❝♦❧✐♥❡❛r❡s✱ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ♣♦♥t♦ D✱ ❜❛st❛ t♦♠❛r

s♦❜r❡ ❛ r❡t❛ AB✱ ❝♦♠ ♠❡s♠❛ ❞✐r❡çã♦ ❞♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦AB✱ ✉♠ ♣♦♥t♦D t❛❧ q✉❡ d(C, D) = d(A, B)✳

✭✐✐✮ ❆❣♦r❛✱ s❡ ✱ ❡ ♥ã♦ ❢♦r❡♠ ❝♦❧✐♥❡❛r❡s✱ ♣❡❧❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ t❡♠♦s q✉❡ ♣♦r ✉♠❛ r❡t❛ ❡ ✉♠ ♣♦♥t♦ ❢♦r❛ ❞❡❧❛ ♣❛ss❛ ✉♠❛ ú♥✐❝❛ r❡t❛ ♣❛r❛❧❡❧❛✳ ❖✉ s❡❥❛✱ ♣♦r C ♣❛ss❛

✉♠❛ ú♥✐❝❛ r❡t❛ ♣❛r❛❧❡❧❛ à r❡t❛AB✳ P♦rt❛♥t♦✱ ❜❛st❛ t♦♠❛r ✉♠ ♣♦♥t♦D s♦❜r❡ ❡ss❛ r❡t❛

♣❛r❛❧❡❧❛ ❛AB✱ ❞❡ ♠♦❞♦ q✉❡ d(C, D) =d(A, B) ❡ AC_∩BD=_∅✳

❈♦♠♣r❡❡♥❞✐❞❛ ❛ ❞❡✜♥✐çã♦ ❞❡ s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡q✉✐♣♦❧❡♥t❡s✱ tr❛③❡♠♦s ❛ s❡✲ ❣✉✐♥t❡ ❞❡✜♥✐çã♦ ❞❡ ✈❡t♦r✳

❉❡✜♥✐çã♦ ✻✳ ❖ ✈❡t♦r~v=−→AB é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❡q✉✐♣♦✲

❧❡♥t❡s ❛♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦AB✱ t❛❧ q✉❡✿

✭✐✮ ❉❛❞♦ ✉♠ ♣♦♥t♦P q✉❛❧q✉❡r✱ ♦ ✈❡t♦r−→P P =−→0 é ✉♠ ✈❡t♦r ♥✉❧♦✳

✭✐✐✮ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ✈❡t♦r ❡ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✷✱ ❞❛❞♦ ✉♠ ✈❡t♦r~v = −→AB ❡ ✉♠ ♣♦♥t♦ C = (x, y)✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♣♦♥t♦D t❛❧ q✉❡ ~v=−−→CD✳

✭✐✐✐✮ ❖ ❝♦♠♣r✐♠❡♥t♦✱ ♦✉ ♠ó❞✉❧♦✱ ❞❡ ✉♠ ✈❡t♦r ~v = (α, β) é ✐♥❞✐❝❛❞♦ ♣♦r d(A, B) =

k~v_k=pα2_+β2✳

❉❡✜♥✐çã♦ ✼✳ ❉❛❞♦ ~v = −→AB✱ ❡♠ q✉❡ A = (a1, a2) ❡ B = (b1, b2)✱ ❞✐③❡♠♦s q✉❡ ♦s

♥ú♠❡r♦sα=b1₋a1❡β =b2₋a2sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈❡t♦r~v✳ ❋❛③❡♥❞♦ ♦ ♣♦♥t♦ ✐♥✐❝✐❛❧

❝♦✐♥❝✐❞✐r ❝♦♠ ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ t❡♠✲s❡ q✉❡~v =−→OP = (0 +α,0 +β)✱

✐st♦ é✱P = (α, β)✳

❊①❡♠♣❧♦ ✶✶✳ ❉❛❞♦s ♦s ♣♦♥t♦s A = (₋2,4)✱ B = (1,1) ❡ C = (₋4,₋1)✱ ❡ ♦ ✈❡t♦r

~v =−→AB✱ ❞❡t❡r♠✐♥❡✿

✭❛✮ P✱ ❞❡ ♠♦❞♦ q✉❡ ~v=−→OP✱ ❝♦♠O = (0,0)✳

❙♦❧✉çã♦✿ ❈♦♠♦ ~v =−→AB✱ ❡♥tã♦ ~v = (1₋(₋2),1₋4) = (0₋3,0₋(₋3)) = −→OP✱

♣♦rt❛♥t♦ P = (3,₋3)✳

✭❜✮ ❖ ♣♦♥t♦ D✱ t❛❧ q✉❡~v =−−→CD✳

❙♦❧✉çã♦✿ ❙❡❥❛ (d1, d2) =D✳ ❋❛③❡♥❞♦~v =−−→CD✱ t❡♠✲s❡✿ ~v=−−→CD =−→AB

⇔(d1₋(₋4), d2₋(₋1)) = (1₋(₋2),1₋4)

⇔(d1+ 4, d2+ 1) = (3,₋3)

⇔(d1, d2) = (−1,−4)

P♦rt❛♥t♦ D= (−1,−4)✳

✭❝✮ ❖ ♠ó❞✉❧♦ ❞❡ ~v✳

❙♦❧✉çã♦✿ ~v= (3,−3)✱ ♣♦rt❛♥t♦ k~vk=p32_{+ (−3)}2 ₌√_{18 = 3}√₂✳

❉❛❞♦ ✉♠ ✈❡t♦r ~v = −→AB ❡ ✉♠ ♣♦♥t♦ C ❞♦ ♣❧❛♥♦✱ ❞❡t❡r♠✐♥❛r ✉♠ ♣♦♥t♦ D t❛❧ q✉❡ CD s❡❥❛ ✉♠ r❡♣r❡s❡♥t❛♥t❡ ❞♦ ✈❡t♦r ~v ♥♦s tr❛♥s♠✐t❡ ❛ ✐❞❡✐❛ ❞❡ q✉❡ ♦ ♣♦♥t♦ C ❡stá

s❡♥❞♦ ❧❡✈❛❞♦ ❛té ♦ ♣♦♥t♦ D✱ ❝❛r❛❝t❡r✐③❛♥❞♦ ❛ tr❛♥❧❛çã♦ ❞❡ ✉♠ ♣♦♥t♦ ♥♦ ♣❧❛♥♦✱ q✉❡

s❡rá ❛❜♦r❞❛❞❛ ❛ s❡❣✉✐r✳

✷✳✶ ❚r❛♥s❧❛çã♦

❆ tr❛♥s❧❛çã♦ Tv✱ ❞❡t❡r♠✐♥❛❞❛ ♣♦r~v = (α, β)❡♠ ✉♠ ♣❧❛♥♦ Π é ✉♠❛ tr❛♥s❢♦r♠❛çã♦

❡♠ q✉❡✱ ♣❛r❛ ❝❛❞❛ ♣♦♥t♦P = (x, y) ❡♠ Π✱ ❢❛③ ❝♦rr❡s♣♦♥❞❡r ✉♠ ♣♦♥t♦ Tv(P) = P′ =

(x+α, y+β)✳

Tv : Π−→Π P _7−→Tv(P).

❖❜s❡r✈❡ q✉❡ −−→P P′ é ✉♠ r❡♣r❡s❡♥t❛♥t❡ ❞♦ ✈❡t♦r_~v✱ ✐st♦ é✱ −−→_{P P}′ _{= (α, β)}✳

Pr♦♣♦s✐çã♦ ✸✳ ❙❡A= (a1, a2)❡B = (b1, b2)✱ ❝♦♠A′ ❡_B′ s✉❛s r❡s♣❡❝t✐✈❛s ✐♠❛❣❡♥s ♦❜✲ t✐❞❛s ♣❡❧❛ tr❛♥s❧❛çã♦Tv✱ ❞❡t❡r♠✐♥❛❞❛ ♣♦r~v = (α, β) ❡♠ Π✱ ❡♥tã♦ d(A, B) =d(A′, B′)✳

❉❡♠♦♥str❛çã♦✿ P♦r ❞❡✜♥✐çã♦✱ t❡♠♦s q✉❡A′ _{= (a1+α, a2+β)}❡_B′ _{= (b1+α, b2+β)}✳ ❊♥tã♦✱

d(A′_{, B}′_{) =} p_{(b1+α−a1−α)}2_{+ (b2+α−a2−α)}2

= p(b1₋a1)2_{+ (b2−a2)}2

= d(A, B).✷