Trong không gian ( Oxyz) , cho mặt phẳng (P) đi qua điểm M(1;2;3) và cắt các trục Ox,Oy,Oz lần lượt tại các điểm A,B ,C . Viết phương trình mặt phẳng (P) sao cho M là trực tâm của tam giác ABC .

A. \( \frac{x}{1} + \frac{y}{2} + \frac{z}{3} = 3\)

B. 6x + 3y - 2z - 6 = 0

C. x + 2y + 3z - 14 = 0

D. x + 2y + 3z - 11 = 0

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Chủ đề: Đề thi THPT QG

Môn: Toán

Lời giải:

Báo sai

Gọi A ( a; 0;0),B ( 0; b; 0) và C ( 0;0;c) với \( abc \ne 0\).

Phương trình mặt phẳng (P) đi qua ba điểm A,B,C là \(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\)

Vì \( M\left( {1\,;\,2\,;\,3} \right) \in \left( P \right)\) nên ta có: \( \frac{1}{a} + \frac{2}{b} + \frac{3}{c} = 1\).

Điểm M là trực tâm của \( \Delta ABC \Leftrightarrow \left\{ \begin{array}{l} AM \bot BC\\ BM \bot AC \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} \overrightarrow {AM} \,.\,\overrightarrow {BC} = 0\\ \overrightarrow {BM} \,.\,\overrightarrow {AC} = 0 \end{array} \right.\).

Ta có: \(\overrightarrow {AM} = \left( {1 - a\,;\,2\,;\,3} \right); % MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8Haaeaaca % WGcbGaam4qaaGaay51GaGaeyypa0ZaaeWaaeaacaaIWaGaaGPaVlaa % cUdacaaMc8UaeyOeI0IaamOyaiaaykW7caGG7aGaaGPaVlaadogaai % aawIcacaGLPaaaaaa!46E6! \overrightarrow {BC} = \left( {0\,;\, - b\,;\,c} \right)\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8Haaeaaca % WGcbGaamytaaGaay51GaGaeyypa0ZaaeWaaeaacaaIXaGaaGPaVlaa % cUdacaaMc8UaaGOmaiabgkHiTiaadkgacaaMc8Uaai4oaiaaykW7ca % aIZaaacaGLOaGaayzkaaaaaa!4782! ;\overrightarrow {BM} = \left( {1\,;\,2 - b\,;\,3} \right)\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8Haaeaaca % WGbbGaam4qaaGaay51GaGaeyypa0ZaaeWaaeaacqGHsislcaWGHbGa % aGPaVlaacUdacaaMc8UaaGimaiaaykW7caGG7aGaaGPaVlaadogaai % aawIcacaGLPaaaaaa!46E4! ;\overrightarrow {AC} = \left( { - a\,;\,0\,;\,c} \right)\)

Ta có hệ phương trình:\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe % qaaiabgkHiTiaaikdacaWGIbGaey4kaSIaaG4maiaadogacqGH9aqp % caaIWaaabaGaeyOeI0IaamyyaiabgUcaRiaaiodacaWGJbGaeyypa0 % JaaGimaaqaamaalaaabaGaaGymaaqaaiaadggaaaGaey4kaSYaaSaa % aeaacaaIYaaabaGaamOyaaaacqGHRaWkdaWcaaqaaiaaiodaaeaaca % WGJbaaaiabg2da9iaaigdaaaGaay5EaaGaeyi1HS9aaiqaaqaabeqa % aiaadkgacqGH9aqpdaWcaaqaaiaaiodaaeaacaaIYaaaaiaadogaae % aacaWGHbGaeyypa0JaaG4maiaadogaaeaadaWcaaqaaiaaigdaaeaa % caaIZaGaam4yaaaacqGHRaWkdaWcaaqaaiaaikdaaeaadaWcaaqaai % aaiodaaeaacaaIYaaaaiaadogaaaGaey4kaSYaaSaaaeaacaaIZaaa % baGaam4yaaaacqGH9aqpcaaIXaaaaiaawUhaaaaa!62F9! \left\{ \begin{array}{l} - 2b + 3c = 0\\ - a + 3c = 0\\ \frac{1}{a} + \frac{2}{b} + \frac{3}{c} = 1 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} b = \frac{3}{2}c\\ a = 3c\\ \frac{1}{{3c}} + \frac{2}{{\frac{3}{2}c}} + \frac{3}{c} = 1 \end{array} \right.\) \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe % qaaiabgkHiTiaaikdacaWGIbGaey4kaSIaaG4maiaadogacqGH9aqp % caaIWaaabaGaeyOeI0IaamyyaiabgUcaRiaaiodacaWGJbGaeyypa0 % JaaGimaaqaamaalaaabaGaaGymaaqaaiaadggaaaGaey4kaSYaaSaa % aeaacaaIYaaabaGaamOyaaaacqGHRaWkdaWcaaqaaiaaiodaaeaaca % WGJbaaaiabg2da9iaaigdaaaGaay5EaaGaeyi1HS9aaiqaaqaabeqa % aiaadkgacqGH9aqpdaWcaaqaaiaaiodaaeaacaaIYaaaaiaadogaae % aacaWGHbGaeyypa0JaaG4maiaadogaaeaadaWcaaqaaiaaigdaaeaa % caaIZaGaam4yaaaacqGHRaWkdaWcaaqaaiaaikdaaeaadaWcaaqaai % aaiodaaeaacaaIYaaaaiaadogaaaGaey4kaSYaaSaaaeaacaaIZaaa % baGaam4yaaaacqGH9aqpcaaIXaaaaiaawUhaaaaa!62F9! \left\{ \begin{array}{l} - 2b + 3c = 0\\ - a + 3c = 0\\ \frac{1}{a} + \frac{2}{b} + \frac{3}{c} = 1 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} b = \frac{3}{2}c\\ a = 3c\\ \frac{1}{{3c}} + \frac{2}{{\frac{3}{2}c}} + \frac{3}{c} = 1 \end{array} \right.\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HS9aai % qaaqaabeqaaiaadggacqGH9aqpcaaIXaGaaGinaaqaaiaadkgacqGH % 9aqpcaaI3aaabaGaam4yaiabg2da9maalaaabaGaaGymaiaaisdaae % aacaaIZaaaaaaacaGL7baaaaa!43B9! \Leftrightarrow \left\{ \begin{array}{l} a = 14\\ b = 7\\ c = \frac{{14}}{3} \end{array} \right.\)

Phương trình mặt phẳng (P) là \( \frac{x}{{14}} + \frac{y}{7} + \frac{{3z}}{{14}} = 1\)\(\Leftrightarrow x + 2y + 3z - 14 = 0\)

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