Phương pháp tọa độ trong giải toán trung học phổ thông

✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❚■➊❯ ❚❍➚ ❍➬◆● ❚❍Õ❨ P❍×❒◆● P❍⑩P ❚➴❆ ✣❐ ❚❘❖◆● ●■❷■ ❚❖⑩◆ ❚❘❯◆● ❍➴❈ P❍✃ ❚❍➷◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ P❍×❒◆● P❍⑩P ❚❖⑩◆ ❙❒ ❈❻P ▼➣ sè✿ ✻✵ ✹✻ ✵✶ ✶✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ữớ ữợ ì ◗❯➮❈ ❚❯❨➎◆ ✣➔ ◆➤♥❣ ✲ ✷✵✶✽ ▲❮■ ❈❆▼ ✣❖❆◆ ❈→❝ ❦➳t q✉↔ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ ❝æ♥❣ tr➻♥❤ ự r tổ ữủ t ữợ sỹ ữợ ữỡ ố t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❝ỉ♥❣ ❜è tr♦♥❣ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ❝õ❛ ♥❣÷í✐ ❦❤→❝✳ ❚ỉ✐ ①✐♥ ❝❤à✉ tr→❝❤ ♥❤✐➺♠ ✈ỵ✐ ♥❤ú♥❣ ❧í✐ ❝❛♠ ✤♦❛♥ ❝õ❛ ♠➻♥❤✳ ❚→❝ ❣✐↔ ỗ ỡ ✤➛✉ t✐➯♥ ❝♦♥ ①✐♥ ❝↔♠ ì♥ ❜❛ ♠➭ ✤➣ s✐♥❤ ❝♦♥ r❛✱ ❜❛♥ t➦♥❣ ❝❤♦ ❝♦♥ ❝✉ë❝ sè♥❣ ❤æ♠ ♥❛②✳ ❈↔♠ ì♥ ❜❛ ♠➭ ✤➣ t✐♥ t÷ð♥❣✱ t❤÷ì♥❣ ②➯✉ ✈➔ tứ trữợ ổ s t ❞÷ï♥❣ ❞ư❝ ❝õ❛ ❜❛ ♠➭ ❝♦♥ s➩ ♠➣✐ ❦❤➢❝ ❣❤✐ ✈➔ ①❡♠ ✤â ❧➔ ✤ë♥❣ ❧ü❝ ✤➸ ❝♦♥ ♣❤➜♥ ✤➜✉ tr ữớ trữợ sỹ trồ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ❡♠ ①✐♥ ❣û✐ ✤➳♥ ❚❤➛② ❣✐→♦ ❚❙✳ ▲÷ì♥❣ ◗✉è❝ ❚✉②➸♥ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ sỹ ữợ t t t❤➛② tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❡♠ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ✤➳♥ ❝→❝ ❚❤➛②✱ ❈ỉ ❣✐→♦ ✈➔ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❑❤♦❛ ❚♦→♥✱ ❚r÷í♥❣ ✣↕✐ ❍å❝ ❙÷ P❤↕♠ ✲ ✣↕✐ ❍å❝ ✣➔ ◆➤♥❣ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ t↕♦ ♥❤✐➲✉ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ✤➸ ❡♠ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❈↔♠ ì♥ ❝→❝ ❛♥❤✱ ❝❤à ✈➔ tr ợ Pữỡ P ỡ ❈➜♣ ❑❤â❛ ✸✷ ✤➣ ❣✐ó♣ ✤ï ❡♠ r➜t ♥❤✐➲✉ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ❚→❝ ❣✐↔ ❚✐➯✉ ỗ ệ ệ ệ ệ é ❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✸ ✺ ✼ ✶✳✶✳ ❱❡❝t♦r ✈➔ ❝→❝ ♣❤➨♣ t♦→♥ ✈➲ ✈❡❝t♦r ✈➔ ❤➺ trö❝ tå❛ ✤ë ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳ ❱❡❝t♦r✱ ❝→❝ ♣❤➨♣ t♦→♥ ✈➔ ❤➺ trư❝ tå❛ ✤ë tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✳ ✶✺ ✶✳✸✳ P❤÷ì♥❣ ♣❤→♣ t♦↕ ✤ë tr♦♥❣ ♠➦t ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✹✳ P❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ❈❍×❒◆● ✷✳ P❍×❒◆● P❍⑩P ●■❷■ ▼❐❚ ❙➮ ❉❸◆● ❚❖⑩◆ ▲■➊◆ ◗❯❆◆ ✣➌◆ P❍×❒◆● P❍⑩P ❚➴❆ ✣❐ ❚❘❖◆● ▼➄❚ P❍➃◆● ✷✽ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✷✳✶✳ ▼ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤✐➸♠ ✈➔ ✤÷í♥❣ t❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷✳ ❈→❝ ❜➔✐ t♦→♥ ✈➲ t➼♥❤ ❝❤➜t ✤è✐ ①ù♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✸✳ ❇➔✐ t♦→♥ ❝ü❝ trà ❤➻♥❤ ❣✐↔✐ t➼❝❤ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✹✳ ❇➔✐ t♦→♥ ❝â ❝❤ù❛ t❤❛♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ❈❍×❒◆● ✸✳ ▼❐❚ ❱⑨■ ⑩P ❉Ư◆● ❈Õ❆ P❍×❒◆● P❍⑩P ❚➴❆ ✣❐ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ✺✾ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✸✳✶✳ ❇➔✐ t♦→♥ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✳ ✳ ✳ ✺✾ ✸✳✷✳ ▼ët sè ❞➜✉ ❤✐➺✉ ♥❤➟♥ ❜✐➳t ❜➔✐ t♦→♥ ❤➻♥❤ ❤å❝ ❦❤æ♥❣ ❣✐❛♥ ❣✐↔✐ ✤÷đ❝ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữợ t ổ ❣✐❛♥ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✸✳✹✳ ▼ët sè ❝→❝❤ ✤➦t ❤➺ trö❝ tồ ợ ởt số t tữớ ũ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✹ ✸✳✺✳ ▼ët sè ❜➔✐ t♦→♥ ❤➻♥❤ ❤å❝ ❦❤æ♥❣ ❣✐❛♥ ❣✐↔✐ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✸✳✻✳ Ù♥❣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝✱ t➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t✱ ♥❤ä ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✸✳✼✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❜❛ ➞♥ t❤ü❝ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë ✽✺ ❑➌❚ ▲❯❾◆ ❱⑨ ❑■➌◆ ◆●❍➚ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✾✸ ✾✹ ✺ ▼Ð ✣❺❯ ✶✳ ỵ t ởt tr ♥❤ú♥❣ ♣❤➙♥ ♥❤→♥❤ q✉❛♥ trå♥❣ ❝õ❛ t♦→♥ ❤å❝ ❧✐➯♥ q✉❛♥ tữợ ✈à tr➼ t÷ì♥❣ ✤è✐ ❝õ❛ ❝→❝ ✈➟t t❤➸ ❝ơ♥❣ ♥❤÷ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥✳ ◆â ❧➔ ❝ỉ♥❣ ❝ư q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ①➙② ❞ü♥❣ ♥➯♥ ❝→❝ ♥❣➔♥❤ t♦→♥ ❤å❝ ❤✐➺♥ ✤↕✐✳ ❍ì♥ t❤➳ ♥ú❛✱ ❤➻♥❤ ❤å❝ ❣✐ú ✈❛✐ trá r➜t q✉❛♥ trå♥❣ tr♦♥❣ ✤í✐ sè♥❣ ❝♦♥ ♥❣÷í✐ ❝ơ♥❣ ♥❤÷ tr♦♥❣ ❦❤♦❛ ❤å❝ ✈➔ ❦➽ t❤✉➟t✳ ❈❤➼♥❤ ✈➻ ❧➩ ✤â✱ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ ❤➻♥❤ ❤å❝ ♥❣➔② ❝➔♥❣ ✤÷đ❝ ❝❤ó trå♥❣ ❤ì♥ ✈➔ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ s➙✉ ❤ì♥ tr♦♥❣ tr÷í♥❣ ❚❍P❚✳ ❚r♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ð tr÷í♥❣ ❚r✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣✱ ❤➻♥❤ ❤å❝ ❦❤æ♥❣ ❝❤➾ ❣✐ú ✈❛✐ trá✱ ✈à tr➼ q✉❛♥ trå♥❣ ♠➔ ♥â ❝á♥ ❧➔ ♠ët ♣❤➛♥ ❦✐➳♥ t❤ù❝ ✈ỉ ❝ị♥❣ ❤➜♣ ❞➝♥ ✈➔ t❤ó ✈à ❞➔♥❤ ❝❤♦ ❤å❝ s✐♥❤✳ ❚✉② ♥❤✐➯♥✱ ❦❤æ♥❣ ➼t ❤å❝ s✐♥❤ ❝↔♠ t❤➜② r➡♥❣ ✤➙② ❧➔ ♠ët ♠æ♥ ❤å❝ q✉→ ❦❤â✱ q✉→ trø✉ t÷đ♥❣✳ ❉♦ ✤â✱ ❝→❝ ❡♠ r➜t ❡ ♥❣↕✐ ❦❤✐ t✐➳♣ ❝➟♥ ♠ỉ♥ ❤å❝ ♥➔② ✈➔ ❣✐→♦ ✈✐➯♥ ❝ơ♥❣ ❣➦♣ ❦❤ỉ♥❣ ➼t ❦❤â ❦❤➠♥ tr♦♥❣ q✉→ tr➻♥❤ tr✉②➲♥ t↔✐ ❦✐➳♥ t❤ù❝✳ ❇➯♥ ❝↕♥❤ ✤â✱ ❤➻♥❤ ❤å❝ ❣✐↔✐ t➼❝❤ ❧➔ ❝ỉ♥❣ ❝ư ❤ú✉ ➼❝❤ ❣✐ó♣ ❤å❝ s✐♥❤ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❤➻♥❤ ❤å❝ ❦❤æ♥❣ ❣✐❛♥ ❤✐➺✉ q✉↔ ✈➔ ❝→❝ ❡♠ ❞➵ ❞➔♥❣ t ỡ ỳ ỵ ữ tr ũ ợ sỹ ữợ t ữỡ ố ❚✉②➸♥✱ ❝❤ó♥❣ tỉ✐ q✉②➳t ✤à♥❤ ❝❤å♥ ✤➲ t➔✐✿ ✏P❍×❒◆● P❍⑩P ❚➴❆ ✣❐ ❚❘❖◆● ●■❷■ ❚❖⑩◆ ❚❘❯◆● ❍➴❈ P❍✃ ❚❍➷◆●✳✑ ✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ▼ö❝ t✐➯✉ ❝õ❛ ✤➲ t➔✐ ♥❤➡♠ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❧➔♠ rã ❝→❝ ✈➜♥ ✤➲✿ P❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë tr♦♥❣ ♠➦t ♣❤➥♥❣✱ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈➔ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥❀ ❝→❝ ❞➜✉ t ữợ t ❤å❝ ❦❤ỉ♥❣ ❣✐❛♥ ❣✐↔✐ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë✳ ✸✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ✻ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ❧➔ ✤✐➸♠✱ ✈❡❝t♦r✱ ❤➺ trư❝ tå❛ ✤ë✱ ✤÷í♥❣ t❤➥♥❣✱ ♠➦t ♣❤➥♥❣ tr♦♥❣ ❤➻♥❤ ❤å❝ tå❛ ✤ë✳ ✹✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ✈➔ tå❛ ✤ë ❦❤ỉ♥❣ ❣✐❛♥ ð ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ❚r✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣✳ ✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❞ü❛ tr➯♥ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣✿ ✲ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ s→❝❤ ❣✐→♦ ✈✐➯♥✱ s→❝❤ ❣✐→♦ ❦❤♦❛ ✈➔ ♠↕♥❣ ✐♥t❡r♥❡t ✈➲ ❝→❝ ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥✳ ✲ P❤➙♥ t➼❝❤✱ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➔✐ ❧✐➺✉ ✤➣ ❝❤å♥ ❧å❝✳ ✲ ❚r❛♦ ✤ê✐ t❤↔♦ ❧✉➟♥ ợ t ữợ ị tỹ t t t õ ỵ tỹ t t ỵ tt t ❧✐➺✉ t❤❛♠ ❦❤↔♦ tèt ❞➔♥❤ ❝❤♦ ❤å❝ s✐♥❤ ❚r✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣✳ ✼✳ ❈➜✉ tró❝ ❧✉➟♥ ✈➠♥ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉ ✈➔ ❦➳t ❧✉➟♥✱ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❧✉➟♥ ✈➠♥ ỗ ữỡ ữỡ tự ỡ s r ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ ✈❡❝t♦r✱ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❤➺ trö❝ tå❛ ✤ë tr♦♥❣ ♠➦t ♣❤➥♥❣ ✈➔ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥✳ ❈❤÷ì♥❣ ■■✿ P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♠ët sè ❞↕♥❣ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë✳ ❚r➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ✈➔ ❧í✐ ❣✐↔✐ ❝❤✐ t✐➳t ♠ët sè ❜➔✐ t♦→♥ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ tå❛ ✤ë tr♦♥❣ ♠➦t ♣❤➥♥❣ ✈➔ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ❚r✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣✳ ❈❤÷ì♥❣ ■■■✿ ▼ët ✈➔✐ →♣ ❞ư♥❣ ữỡ tồ tr ổ ì ✶ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ❈→❝ ❡♠ ✤➣ ✤÷đ❝ ❜✐➳t tợ t tr t ữỡ tr➻♥❤ ❤➻♥❤ ✶✵✳ ❚ỵ✐ ❦➻ ✷ ❧ỵ♣ ✶✷ ❝→❝ ❡♠ ✤➣ ✤÷đ❝ ❤å❝ ♠ð rë♥❣ ❤ì♥ s❛♥❣ ❦❤ỉ♥❣ ❣✐❛♥✳ ❙➩ ❝â ♥❤ú♥❣ ❜➔✐ t➟♣ ♠➔ ❡♠ s➩ t❤➯♠ ❝→❝❤ ❣✐↔✐ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë✱ ❦❤ỉ♥❣ q✉→ ❦❤â ✈➔ r➜t t❤ó ✈à✳ ◆❤÷♥❣ ✤➸ ❣✐↔✐ ✤÷đ❝ ❝→❝ ❜➔✐ t♦→♥ t❤ó trữợ t ỳ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ tå❛ ✤ë s❛✉ ✤➙②✳ ✶✳✶✳ ❱❡❝t♦r ✈➔ ❝→❝ ♣❤➨♣ t♦→♥ ✈➲ ✈❡❝t♦r ✈➔ ❤➺ trö❝ tå❛ ✤ë ❆✳ ❱❡❝t♦r tr♦♥❣ ♠➦t ♣❤➥♥❣ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❱❡❝t♦r ởt t õ ữợ ã ởt ố ỏ ữợ tr ã t ữủ tr ã ữợ tứ ố ữủ ã ữớ t q ố ✈➔ ✤✐➸♠ ♥❣å♥ ✤÷đ❝ ❣å✐ ❧➔ ✈❡❝t♦r✳ ❣✐→ ❝õ❛ ❍➻♥❤✳ ỵ tr õ ố AB A ữủ ỵ AB B ữủ ỵ ❧➔ ❤❛② −→ AB ✱ ✈➔ ✤ë ❞➔✐ ❝õ❛ ✈❡❝t♦r AB ởt tr ỏ ữủ ỵ ởt ❝❤ú ❝→✐ ✐♥ t❤÷í♥❣ ♣❤➼❛ tr➯♥ ❝â ♠ơ✐ t➯♥ ♥❤÷ → − − → → − − a✱ b✱ → c✱− u✱ → v ✤ë ❞➔✐ ❝õ❛ → − a ữủ ỵ | a | ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❱❡❝t♦r ❦❤æ♥❣ ❧➔ ✈❡❝t♦r ❝â ✤✐➸♠ ✤➛✉ ✈➔ ✤✐➸♠ ❝✉è✐ trị♥❣ ♥❤❛✉✱ ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ ữ ã ố trị♥❣ ♥❤❛✉✳ • ✣ë ❞➔✐ ❜➡♥❣ ✵✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ❍❛✐ ✈❡❝t♦r ✤÷đ❝ ❣å✐ ❧➔ ❝ị♥❣ ♣❤÷ì♥❣ ♥➳✉ ❝❤ó♥❣ ♥➡♠ tr➯♥ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ ❤♦➦❝ ❝ị♥❣ ♥➡♠ tr➯♥ ởt ữớ t ỵ AB CD ữ AB ú ỵ CD AB CD, A, B, C, D t❤➥♥❣ ❤➔♥❣ ❚❛ ❝â ✶✮ tr ũ ữỡ t ũ ữợ ữủ ữợ tr ổ õ ữỡ ữợ tũ ỵ tr ữủ ữợ ũ ỵ a = b✳ → − → − a = b ⇔ ❜➡♥❣ ♥❤❛✉ ♥➳✉ ❝❤ó♥❣ ❝ị♥❣ ◆❤÷ ✈➟②✱ → − − |→ a|= b , → − → − a b ❇✳ ❈→❝ ♣❤➨♣ t♦→♥ ✈❡❝t♦r ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻✳ ❚ê♥❣ ❝õ❛ ❤❛✐ ✈❡❝t♦r → − a ✈➔ → − b ❧➔ ♠ët tr ữủ ữ s ã ứ ởt tũ ỵ A tr t ỹ tr → B ❞ü♥❣ ✈❡❝t♦r BC = b ✳ −→ ❑❤✐ ✤â ✈❡❝t♦r AC ✤÷đ❝ ❣å✐ ❧➔ ✈❡❝t♦r tê♥❣ − −→ → − → ✈✐➳t AB = a + b ✳ • −→ → AB = − a✳ ❚ø ✤✐➸♠ ❝õ❛ ❤❛✐ ✈❡❝t♦r → − a ✈➔ → − b, ✈➔ t❛ ✾ → − a → − a → − b ❍➻♥❤✳ ✶✳✷✳ → − b → − → − a + b ❚ê♥❣ ❝õ❛ ❤❛✐ ✈❡❝t♦r → − − − ❚➼♥❤ ❝❤➜t ✶✳✶✳✼✳ ❈❤♦ ❜❛ ✈❡❝t♦r → a, b, → c , t❛ ❝â ✶✮ ❚➼♥❤ ❝❤➜t ❣✐❛♦ ❤♦→♥✿ → − → − − → − a + b = b +→ a ✷✮ ❚➼♥❤ ❝❤➜t ❦➳t ❤ñ♣✿ → − − → − → − −c = → − c a + b +→ a + b +→ ✸✮ ❚➼♥❤ ❝❤➜t ❝õ❛ ✈❡❝t♦r ❦❤æ♥❣✿ → − → − − → − − a + = +→ a =→ a ✹✮ ◗✉② t➢❝ ✸ ✤✐➸♠✿ ❈❤♦ ✸ ✤✐➸♠ A✱ B ✱ C ❜➜t ❦ý✳ ❑❤✐ ✤â✱ −→ −−→ −→ AB + BC = AC B C A ❍➻♥❤✳ ✶✳✸ ✺✮ ◗✉② t➢❝ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤✿ −→ −−→ −→ AB + AD = AC ✽✶ z S M y D N A O C B x ❍➻♥❤✳ ✸✳✶✹ √ B(a; 0; 0)✱ A(0; a 3; 0)✱ S(0; 0; m) ✈ỵ✐ m > 0✳ (SAB) ❧➔ x y z + √ + = a a m ❚❛ ❝â tr➻♥❤ ❉♦ ✤â✱ t❛ ❝â √ a d(O, (SAB)) = √ a | − 1| ⇔ = 1 + + a2 3a2 m2 + = 16 ⇔3a2 3a2 m2 a2 ⇔m = a ⇔m = (m > 0) ❙✉② r❛ ♣❤÷ì♥❣ ✽✷ ◆❤÷ ✈➟②✱ ❚❛ ❝â SS.ABCD = SO · SABCD √ √ a a = · · 2a2 = 3 √ √ √ a a a a M 0; ; , D(−a; 0; 0), C(0; −a 3; 0), N ; ;0 , 2 √ √ −−→ −−→ 3a a a a ; , BM − a; ; BN − ; − 2 √ √ −−→ −−→ a2 3a2 5a2 ⇒ [BN , BM ] = − ; ;− 8 ❉♦ ✈➟②✱ √ −−→ −−→ 26 |[BN , BM ]| d(M, BN ) = = −−→ |BN | ❱➼ ❞ö ✸✳✺✳✻✳ S.ABCD ❝â ✤→② ABCD ❧➔ ❤➻♥❤ t❤♦✐ ❝â ◦ t➙♠ O ❝↕♥❤ ❜➡♥❣ 2a✱ BAD = 60 ✳ ●å✐ M ✱ N ❧➛♥ ❧÷đt ❧➔ tr✉♥❣ ✤✐➸♠ CD✱ AD✳ ❍➻♥❤ ❝❤✐➳✉ ❝õ❛ ✤➾♥❤ S ❧➯♥ (ABCD) ❧➔ tr✉♥❣ ✤✐➸♠ H ❝õ❛ AN ✱ SH = a✳ ❚➼♥❤ t❤➸ t➼❝❤ ❦❤è✐ ❝❤â♣ S.ABM D ✈➔ ❝♦s✐♥ ❝õ❛ ❣â❝ ❣✐ú❛ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ SN ✈➔ BM ✳ ❈❤♦ ❤➻♥❤ ❝❤â♣ ●✐↔✐✳ ●å✐ DK ❧➔ ✤÷í♥❣ ❝❛♦ ❝õ❛ ❤➻♥❤ t❤❛♥❣ ABM D✱ ✈➻ ♥➯♥ ∆BDA ❙✉② r❛ ◆❤÷ ✈➟②✱ ✤➲✉✳ ❉♦ AB = AD = 2av BAD = 60◦ √ √ ✤â✱ AO = a 3✱ BD = 2a✱ OB = a✱ DK = a 3✳ √ (DM + AB) · DK (2a + a) · a SABM D = = 2 √ 3a2 = t❤➸ t➼❝❤ ❝õ❛ ❦❤è✐ ❝❤â♣ S.ABM D ❧➔ √ a3 VS.ABM D = SH · SABM D = ✽✸ ❜✳ ❈❤å♥ ❤➺ trö❝ tå❛ ✤ë Oxyz ữ ợ B Ox A Oy ✱ S ∈ Oz ✳ z S y D N M A H K O C B x ❍➻♥❤✳ ✸✳✶✺ ❚❛ ❝â √ a a B(a; 0; 0), A(0; a 3; 0), D(−a; 0; 0), N − ; ;0 , 2 √ √ √ a 3a a 3a a a H − ; ;0 ,S − ; ;a ,M − ;− ;0 , 4 √ 4 2 √ −→ −−→ a a 3a a ; −a , BM − ; − ;0 SN − ; − 4 2 √ −→ −−→ 15 ✈➟②✱ cos(SN, BM ) = | cos(SN , BM )| = ✳ 10 √ ❉♦ ✸✳✻✳ Ù♥❣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝✱ t➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t✱ ♥❤ä ♥❤➜t ❱➼ ❞ö ✸✳✻✳✶✳ x y z≥− x+y+z = √ ❈❤♦ ❝→❝ sè t❤ü❝ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ √ √ ✱ ✱ √ t❤ä❛ ♠➣♥ ✳ 5x + + 5y + + 5z + ≤ 7✳ ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ♥➔♦❄ √ − − ●✐↔✐✳ ❳➨t ❤❛✐ ✈❡❝tì → u (1; 1; 1), → v( 5x + 2; √ 5y + 2; √ 5z + 2) ✽✹ → − − − − u ·→ v ≤ |→ u | · |→ v |✱ t❛ ❝â √ √ 5x + + 5y + + 5z + √ ≤ · 5x + + 5y + + 5z + √ = · 5(x + y + z) + √ √ √ √ √ = · · + = · 21 = ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ → − − u✱ → v ❧➔ tỡ ũ ữợ 5x + = 5y + = 5z + x+y+z =3 ❱➼ ❞ö ✸✳✻✳✷✳ ❈❤♦ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ ⇔ x = y = z = a✱ b✱ c t❤ä❛ ♠➣♥ a2 + 3b2 + 5c2 = ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ 2a + b + 3c ≤ ●✐↔✐✳ 276 √ √ − → − v 2; √ ; √ u (a; b 3; c 5), → → − → − → − → − u · v ≤ | u | · | v |✱ t❛ ❝â ❳➨t ❤❛✐ ✈❡❝tì ✤➥♥❣ t❤ù❝ 2a + b + 3c ≤ = √ √ a2 + 3b2 + 5c2 · 92 = 15 9· 22 + √ + ⑩♣ ❞ö♥❣ ❜➜t √ 276 ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❱➼ ❞ư ✸✳✻✳✸✳ ❈❤♦ ❜❛ sè t❤ü❝ ❧ỵ♥ ♥❤➜t✱ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ x✱ y ✱ z t❤ä❛ x2 + y + z = 1✳ F = |2x + 2y − z − 9|✳ ❚➻♠ ❣✐→ trà ●✐↔✐✳ ❳➨t ♠➦t ❝➛✉ (S) : x2 + y2 + z = 1✱ t➙♠ O✱ ❜→♥ ❦➼♥❤ R = ✈➔ ♠➦t ♣❤➥♥❣ (α) : 2x + 2y − z − = 0✳ ✣÷í♥❣ t❤➥♥❣ (∆) q✉❛ O ✈➔ ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐  x = 2t y = 2t  z = −t (α) ❝â ♣❤÷ì♥❣ tr➻♥❤ ✽✺ ❚å❛ ✤ë ❣✐❛♦ ✤✐➸♠ (∆) ✈➔ (S) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤  x = 2t    y = 2t z = −t    x + y2 + z2 = ◆❤÷ ✈➟②✱ ❚❛ ❝â (∆) ❝➢t (S) t↕✐ A ⇔ t2 = 2 ; ;− 3 (A, (α)) = 2✱ d(B, (α)) = 4✳ d(M, (S)) = ✱ 1 ⇔t=± 2 B − ;− ; 3 ❚❛ ❧➜② ✳ M (x; y; z) ∈ (S)✳ |2x + 2y − z − 9| √ = 22 + 2 + ❚❛ ❧✉æ♥ ❝â d(A, (α)) ≤ d(M, (α)) ≤ d(M, (α)) ⇔2≤ F ≤4 ⇔ ≤ F ≤ 12 ❉♦ ✤â✱ 2 x= , y= , z=− , 3 2 F = 12 ❦❤✐ x = − , y = − , z = 3 F = ❦❤✐ ✸✳✼✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❜❛ ➞♥ t❤ü❝ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë ❱➼ ❞ư ✸✳✼✳✶✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤  x + y + z = 3,    x2 + y + z = 3,    x + y + z = (1) (2) (3) ●✐↔✐✳ ▼➦t ❝➛✉ (S) : x2 + y2 + z = 3✱ t➙♠ O✱ ❜→♥ ❦➼♥❤ R = (α) : x + y + z − = t✐➳♣ ①ó❝ ✈ỵ✐ ♥❤❛✉ ✈➻ d(O, (α)) = √ √ | − 3| = = R 11 + 1 + 1 √ ✈➔ ✽✻ ❉♦ ✤â✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ x2 + y + z = x + y + z = 3, ❝â ♥❣❤✐➺♠ ❝❤✉♥❣ ❞✉② ♥❤➜t x=y=1 t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✮✳ ◆❤÷ ✈➟②✱ ❤➺ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❱➼ ❞ö ✸✳✼✳✷✳ x = y = z = 1✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤  3x − 3y − z + = 0, 3x − y + z − = 0,  3x − 4y − 24z = ●✐↔✐✳ ❳➨t ❤➺ (1) (2) (3) 3x − 3y − z + = 0, 3x − y + z − = Oxyz ✱ ❤➺ tr➯♥ ❧➔ ❣✐❛♦ t✉②➳♥ ❝õ❛ (P ) : 3x − 3y − z + = ✈➔ (Q) : 3x − y + z − = 0✳ ❚r♦♥❣ ❤➺ trö❝ tå❛ ✤ë ❤❛✐ ♠➦t ♣❤➥♥❣ (d) ❧➔ ❣✐❛♦ t✉②➳♥ ❝õ❛ (P ) ✈➔ (Q)✳ ❑❤✐ ✤â✱ (d) ❝â ✈❡❝t♦r ❝❤➾ ♣❤÷ì♥❣ → − − → u = [− n→ P , nQ ] = (2; 3; −3)✳ ●✐↔✐ ❤➺ ●å✐ 3x − 3y − z + = 0, 3x − y + z − = ❈❤å♥ x = 0✱ t❛ ❣✐↔✐ ✤÷đ❝ ✤â✱ ✤÷í♥❣ t❤➥♥❣ (d) y=0 ✈➔ z=2 ♥➯♥ (d) ✤✐ q✉❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ t❤❛♠ sè ❧➔  x = 2t y = 3t  z = − 3t ❚❤❛② x = 2t✱ y = 3t✱ z = − 3t ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✮ 3(2t)2 − 4(3t)2 − 24(2 − 3t) = ⇔ −24t2 + 72t − 48 = ⇔ t = ∨ t = A(0; 0; 2)✳ ❑❤✐ ✽✼ ❱ỵ✐ t = 1✱ t❛ ❝â x = 2✱ y = 3✱ z = −1✳ ❱ỵ✐ t = 2✱ t❛ ❝â x = 4✱ y = 6✱ z = −4✳ ❉♦ ✈➟②✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ❤❛✐ ♥❣❤✐➺♠ ❱➼ ❞ö ✸✳✼✳✸✳ (2; 3; −1) ✈➔ (4; 6; −4)✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤  x + 2yz = 1, y + 2xz = 2,  z + 2xy = (1) (2) (3) ●✐↔✐✳ ❈ë♥❣ ✈➳ t❤❡♦ ✈➳ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✮✱ ✭✷✮✱ ✭✸✮ t❛ ✤÷đ❝ (x + y + z)2 = x+y+z =2 ⇔ x + y + z = −2 ❚❤ü❝ ❤✐➺♥ ✭✶✮✲✭✷✮ ✈➳ t❤❡♦ ✈➳✱ t❛ ✤÷đ❝ (x − z)(x − 2y + z) = x − z = 0, ⇔ x − 2y + z = ❑➳t ❤đ♣ ❧↕✐ t❛ ❝â ✹ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉  x + y + z + = x−z =0  x + 2yz − = ✭■✮ x+y+z+2=0 x − 2y + z = x2 + 2yz − = ✭■■✮ x+z+z−2=0 x−z =0 x2 + 2yz − = ✭■■■✮ x+y+z−2=0 x − 2y + z = x2 + 2yz − = ✭■❱✮ ●✐↔✐ ❤➺ ✭■✮✳ ✣÷í♥❣ ❣✐❛♦ t✉②➳♥ x+y+z+2=0 x−z =0 ✽✽ ❝â ❞↕♥❣ t❤❛♠ sè  x = t y = −2 + 2t  z = t, t❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ x2 + 2yz − = t❛ ✤÷đ❝ 3t2 + 4t + = t = −1 ⇔ t=− • ợ ã ợ t = t õ x = −1✱ y = 0✱ z = −1✳ 1 t = − ✱ t❛ ❝â x = − ✱ y = − ✱ z = − ✳ 3 3 ◆❤÷ ✈➟②✱ ❤➺ ✭■✮ ❝â ❤❛✐ ♥❣❤✐➺♠ ❧➔ (−1; 0; −1) ✈➔ − ;− ;− 3 ✳ ●✐↔✐ ✭■■✮✳ ✣÷í♥❣ t❤➥♥❣ ❣✐❛♦ t✉②➳♥ x + y + z + = 0, x − 2y + z = ❝â ❞↕♥❣ t❤❛♠ sè ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤  x=t       y=−      z = − − t x + 2yz − = t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ ✈æ ♥❣❤✐➺♠✳ ❉♦ ✈➟②✱ ❤➺ ✭■■✮ ✈æ ♥❣❤✐➺♠✳ ●✐↔✐ ✭■■■✮✳ ✣÷í♥❣ t❤➥♥❣ ❣✐❛♦ t✉②➳♥ x+y+z−2=0 x−z =0 ❝â ❞↕♥❣ t❤❛♠ sè  x = t y = − 2t  z = t 9t2 + 12t + = 0✱ ✽✾ ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ x2 + 2yz − = t❛ ✤÷đ❝ −3t2 + 4t − = ã ợ ã ợ t=1 t= t = 1✱ t❛ ❝â x = 1✱ y = 0✱ z = 1✳ 1 t = t❛ ❝â x = ✱ y = ✱ z = ✳ 3 3 ◆❤÷ ✈➟②✱ ❤➺ ✭■■■✮ ❝â ✷ ♥❣❤✐➺♠ ❧➔ (1; 0; 1) ✱ ; ; 3 ✳ ●✐↔✐ ✭■❱✮✳ ✣÷í♥❣ t❤➥♥❣ ❣✐❛♦ t✉②➳♥ x+y+z−2=0 x − 2y + z = ❝â ❞↕♥❣ t❤❛♠ sè ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤  x=t       y=      z = − t x + 2yz − = t❛ ✤÷đ❝ 9t2 − 12t + = P❤÷ì♥❣ tr➻♥❤ tr➯♥ ✈æ ♥❣❤✐➺♠✳ ❉♦ ✈➟②✱ ❤➺ ✭■❱✮ ✈æ ♥❣❤✐➺♠✳ ❍➺ ✤➣ ❝❤♦ ❝â ✹ ♥❣❤✐➺♠ 1 , ; ; (−1; 0; −1), (1; 0; 1), − ; − ; − 3 3 3 ❱➼ ❞ư ✸✳✼✳✹✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉  x + y + z = 6, |x − y| + 3z = 4,  xyz = (1) (2) (3) ✾✵ ●✐↔✐✳ ❚❛ ❝â (2) ⇔ |x − y| = − 3z ❙✉② r❛ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ − 3z ≥ (2) ⇔ ✭✯✮ x − y = − 3z x − y = 3z − ❑➳t ❤ñ♣ ❧↕✐ t❛ ❝â ❤❛✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉✿  x + y + z = x − y + 3z =  xyz =  x + y + z = x − y − 3z = −4  xyz = ●✐↔✐ ❤➺ ✭■✮✳ ✣÷í♥❣ ❣✐❛♦ t✉②➳♥ x+y+z−6=0 x − y + 3z − = ❝â ♣❤÷ì♥❣ tr➻♥❤ t❤❛♠ sè ❧➔  x = + 2t y =1−t  z = −t ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ xyz = t❛ ✤÷đ❝ t =  2t3 + 3t2 − 5t − = ⇔  t = −2 t = −1  x=8     t = t❛ ❝â y = − t❤ä❛ ♠➣♥ ✭✯✮✳    z = ã ợ ã ợ t = t õ ã ợ t = −1 t❛ ❝â  x = y=3  z=2  x = y=2  z=1 ◆❤÷ ✈➟②✱ ❤➺ ✭■✮ ❝â ✷ ♥❣❤✐➺♠ ❦❤æ♥❣ t❤ä❛ ♠➣♥ ✭✯✮✳ t❤ä❛ ♠➣♥ ✭✯✮✳ 8; − ; − 2 ✱ (3; 2; 1)✳ ●✐↔✐ ❤➺ ✭■■✮✳ ✣÷í♥❣ t❤➥♥❣ ❣✐❛♦ t✉②➳♥ x+y+z−6=0 x − y − 3z + = ❝â ♣❤÷ì♥❣ tr➻♥❤ t❤❛♠ sè  x = t y = − 2t  z = −1 + t ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ xyz = t❛ ữủ ã ợ ã ợ ã ợ t=3  2t3 − 9t2 + 7t + = ⇔  t = − t =  x = t = t❛ ❝â y = ❦❤æ♥❣ t❤ä❛ ♠➣♥ ✭✯✮✳  z=2    x = −   t = − t❛ ❝â y = t❤ä❛ ♠➣♥ ✭✯✮✳    z = −  x = t = t❛ ❝â y = t❤ä❛ ♠➣♥ ✭✯✮✳  z=1 ◆❤÷ ✈➟②✱ ❤➺ ✭■■✮ ❝â ✷ ♥❣❤✐➺♠ − ; 8; − 2 ✱ (2; 3; 1)✳ ✾✷ ❍➺ ✤➣ ❝❤♦ ❝â ✹ ♥❣❤✐➺♠ 3 , 8; − ; − (3; 2; 1), (2; 3; 1), − ; 8; − 2 2 ✾✸ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✏P❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë tr♦♥❣ ❣✐↔✐ t♦→♥ ❚r✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣✑ ✤➣ ✤↕t ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉✿ ✭❛✮ ❚➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë✳ ✭❜✮ ❍➺ t❤è♥❣ ✈➔ ♣❤➙♥ ❧♦↕✐ ♠ët sè ❞↕♥❣ t♦→♥ tr♦♥❣ ♠➦t ♣❤➥♥❣ ✈➔ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ tr ữỡ tr t r tổ ố ợ ♠é✐ ❞↕♥❣ t♦→♥ ✤➲✉ ❝â ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ✈➔ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ❝ö t❤➸✳ ✣➦❝ ❜✐➺t✱ ❧✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② r➜t t÷í♥❣ ♠✐♥❤ ✈➲ ❝→❝❤ ✈✐➳t ♣❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ t❤➥♥❣✱ ♠➦t ♣❤➥♥❣✱ ❝→❝ ❞➜✉ ❤✐➺✉ ♥❤➟♥ ❜✐➳t ✈➔ ữợ t ổ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë✳ ❚r♦♥❣ s✉èt t❤í✐ ❣✐❛♥ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✱ ♠➦❝ ❞ị ❡♠ ✤➣ r➜t ❝❤ó t➙♠✱ s♦♥❣ ❞♦ sü ❤↕♥ ❝❤➳ ✈➲ ❦❤↔ ♥➠♥❣ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❘➜t ♠♦♥❣ ♥❤➟♥ ữủ sỹ õ ỵ qỵ ổ ❜↕♥ ✤➸ ❧✉➟♥ ✈➠♥ ❝õ❛ ❡♠ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✾✹ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❚✐➳♥❣ ❱✐➺t ❬✶❪ P❤↕♠ ❑❤➢❝ ❇❛♥✱ ❱➠♥ ◆❤÷ ❈÷ì♥❣✱ ▲➯ ❍✉② ❍ị♥❣✱ ❚❛ ▼➙♥✱ ✣♦➔♥ ◗✉ý♥❤ ✭✷✵✶✵✮✱ ✧❍➻♥❤ ❤å❝ ✶✷ ♥➙♥❣ ❝❛♦✧✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❉ö❝✳ ❬✷❪ ◆❣✉②➵♥ ❚➔✐ ❈❤✉♥❣✱ ❍✉ý♥❤ ❱➠♥ ▼✐♥❤ ✭✷✵✶✺✮✱ ✧❇➼ q✉②➳t t✐➳♣ ❝➟♥ ❤✐➺✉ q✉↔ ❦ý t❤✐ ❚❍P❚ q✉è❝ ❣✐❛ ❍➻♥❤ ❤å❝ ❣✐↔✐ t➼❝❤ ❦❤æ♥❣ ❣✐❛♥✧✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✸❪ ❱➠♥ ◆❤÷ ❈÷ì♥❣✱ P❤↕♠ ❱ơ ❑❤➯✱ ❇ị✐ ❱➠♥ ◆❣❤à✱ ✣♦➔♥ ◗✉ý♥❤ ✭✷✵✵✻✮✱ ✧❍➻♥❤ ❤å❝ ✶✵ ♥➙♥❣ ❝❛♦✧✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❉ö❝✳ ❬✹❪ ỗ ự õ ỹ ổ ❧í✐ ❣✐↔✐ ❝❤✐ t✐➳t ❤➻♥❤ ❤å❝ ✶✵✧✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❍å❝ ◗✉è❝ ●✐❛ ❍➔ ◆ë✐✳ ❬✺❪ ◆❣✉②➵♥ ▼ë♥❣ ❍②✱ ◆❣✉②➵♥ ❱➠♥ ✣♦➔♥❤✱ ❚r➛♥ ✣ù❝ ❚❤✉②➯♥ ✭✷✵✶✵✮✱ ✧❇➔✐ t➟♣ ❤➻♥❤ ❤å❝ ✶✵✧✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❉ö❝✳ ❬✻❪ ❱ã ✣↕✐ ▼❛✉ ✭✶✾✾✽✮✱ ✧P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❤➻♥❤ ❤å❝✧✱ ①✉➜t ❜↔♥ ❚r➫✳ ◆❤➔ ❬✼❪ ❚r➛♥ ❚❤➔♥❤ ▼✐♥❤✱ P❤❛♥ ▲÷✉ ❇✐➯♥✱ ❱ô ❱➽♥❤ ❚❤→✐✱ P❤❛♥ ❚❤❛♥❤ ❚❤✐➯♥ ✭✷✵✵✸✮✱ ✧●✐↔✐ t♦→♥ ❤➻♥❤ ❤å❝✧✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❉ö❝✳ ❬✽❪ ✣➦♥❣ ❚❤➔♥❤ ◆❛♠ ✭✷✵✶✹✮✱ ✧◆❤ú♥❣ ✤✐➲✉ ❝➛♥ ❜✐➳t ❧✉②➺♥ t❤✐ ✤↕✐ ❤å❝ ✲ ❑ÿ t❤✉➟t ❣✐↔✐ ♥❤❛♥❤ ❤➻♥❤ ♣❤➥♥❣ ❣✐❛ ❍➔ ◆ë✐✳ Oxy ✧✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ◗✉è❝ ❬✾❪ ▲➯ ỗ ự ổ t t♦→♥ ❤➻♥❤ ❤å❝ ✶✷✧✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ◗✉è❝ ỗ ự ✭✷✵✵✺✮✱ ✧❇ë ✤➲ ❧✉②➺♥ t❤✐ t❤û ❚❍P❚ ◗✉è❝ ❣✐❛ ♠æ♥ ❚♦→♥✧✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ✾✺ ❬✶✶❪ ◆❣✉②➵♥ ❱➠♥ ◆❤♦✱ ◆❣✉②➵♥ ❱➠♥ ❚❤ê ✭✷✵✶✻✮✱ ✧❇ë ✤➲ ❧✉②➺♥ t❤✐ t❤û ❚❍P❚ q✉è❝ ❣✐❛ ♠æ♥ ❚♦→♥✧✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✶✷❪ ❚r➛♥ Pữỡ ỗ ự t ❧✉②➺♥ t❤✐ ✤↕✐ ❤å❝ ♠æ♥ ❚♦→♥ ❤➻♥❤ ❣✐↔✐ t➼❝❤✧✱ ◆❤➔ ①✉➜t ❜↔♥ ❍➔ ◆ë✐✳ ❬✶✸❪ ▲➯ ▼➟✉ ❚❤è♥❣✱ ▲➯ ▼➟✉ Pữỡ t ợ ✶✷✧✱ ◆❤➔ ①✉➜t ❜↔♥ ❚r➫✳