Cho số phức z thay đổi thỏa mãn \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaaca % WG6bGaey4kaSIaaGymaiabgkHiTiaadMgaaiaawEa7caGLiWoacqGH % 9aqpcaaIZaaaaa!3F4F! \left| {z + 1 - i} \right| = 3\). Giá trị nhỏ nhất của biểu thức \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 % da9iaaikdadaabdaqaaiaadQhacqGHsislcaaI0aGaey4kaSIaaGyn % aiaadMgaaiaawEa7caGLiWoacqGHRaWkdaabdaqaaiaadQhacqGHRa % WkcaaIXaGaeyOeI0IaaG4naiaadMgaaiaawEa7caGLiWoaaaa!4A12! A = 2\left| {z - 4 + 5i} \right| + \left| {z + 1 - 7i} \right|\) bằng \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaka % aabaGaamOyaaWcbeaaaaa!37DB! a\sqrt b \)(với a,b là các số nguyên). Tính S = 2a + b?

A. 20

B. 18

C. 23

D. 17

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Chủ đề: Đề thi THPT QG

Môn: Toán

Lời giải:

Báo sai

Đặt \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaiabg2 % da9iaadIhacqGHRaWkcaWG5bGaamyAaiaaykW7caaMc8+aaeWaaeaa % caWG4bGaaiilaiaadMhacqGHiiIZcqWIDesOaiaawIcacaGLPaaaaa % a!4601! z = x + yi\,\,\left( {x,y \in R} \right)\) nên giả thiết \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HS9aae % WaaeaacaWG4bGaey4kaSIaaGymaaGaayjkaiaawMcaamaaCaaaleqa % baGaaGOmaaaakiabgUcaRmaabmaabaGaamyEaiabgkHiTiaaigdaai % aawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaI5aaa % aa!4532! \Leftrightarrow {\left( {x + 1} \right)^2} + {\left( {y - 1} \right)^2} = 9\).

Do đó \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 % da9iaaikdadaGcaaqaamaabmaabaGaamiEaiabgkHiTiaaisdaaiaa % wIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaqadaqaai % aadMhacqGHRaWkcaaI1aaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI % YaaaaaqabaGccqGHRaWkdaGcaaqaamaabmaabaGaamiEaiabgUcaRi % aaigdaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWk % daqadaqaaiaadMhacqGHsislcaaI3aaacaGLOaGaayzkaaWaaWbaaS % qabeaacaaIYaaaaaqabaaaaa!4FB4! A = 2\sqrt {{{\left( {x - 4} \right)}^2} + {{\left( {y + 5} \right)}^2}} + \sqrt {{{\left( {x + 1} \right)}^2} + {{\left( {y - 7} \right)}^2}} \)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaO % aaaeaadaqadaqaaiaaikdacaWG4bGaeyOeI0IaaGioaaGaayjkaiaa % wMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaabmaabaGaaGOmai % aadMhacqGHRaWkcaaIXaGaaGimaaGaayjkaiaawMcaamaaCaaaleqa % baGaaGOmaaaaaeqaaOGaey4kaSYaaOaaaeaadaqadaqaaiaadIhacq % GHRaWkcaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa % ey4kaSYaaeWaaeaacaWG5bGaeyOeI0IaaG4naaGaayjkaiaawMcaam % aaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiodadaWadaqaamaabmaa % baGaamiEaiabgUcaRiaaigdaaiaawIcacaGLPaaadaahaaWcbeqaai % aaikdaaaGccqGHRaWkdaqadaqaaiaadMhacqGHsislcaaIXaaacaGL % OaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGyoaaGaay % 5waiaaw2faaaWcbeaaaaa!60D4! = \sqrt {{{\left( {2x - 8} \right)}^2} + {{\left( {2y + 10} \right)}^2}} + \sqrt {{{\left( {x + 1} \right)}^2} + {{\left( {y - 7} \right)}^2} + 3\left[ {{{\left( {x + 1} \right)}^2} + {{\left( {y - 1} \right)}^2} - 9} \right]} \)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyzIm7aaO % aaaeaadaqadaqaaiaaikdacaWG4bGaeyOeI0IaaGioaiabgkHiTiaa % ikdacaWG4bGaeyOeI0IaaGOmaaGaayjkaiaawMcaamaaCaaaleqaba % GaaGOmaaaakiabgUcaRmaabmaabaGaaGOmaiaadMhacqGHRaWkcaaI % XaGaaGimaiabgkHiTiaaikdacaWG5bGaey4kaSIaaGynaaGaayjkai % aawMcaamaaCaaaleqabaGaaGOmaaaaaeqaaOGaeyypa0JaaGynamaa % kaaabaGaaGymaiaaiodaaSqabaGcdaqadaqaamaakaaabaGaamyyam % aaCaaaleqabaGaaGOmaaaakiabgUcaRiaadkgadaahaaWcbeqaaiaa % ikdaaaaabeaakiabgUcaRmaakaaabaGaam4yamaaCaaaleqabaGaaG % OmaaaakiabgUcaRiaadsgadaahaaWcbeqaaiaaikdaaaaabeaakiab % gwMiZoaakaaabaWaaeWaaeaacaWGHbGaey4kaSIaam4yaaGaayjkai % aawMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaabmaabaGaamOy % aiabgUcaRiaadsgaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaa % aabeaaaOGaayjkaiaawMcaaaaa!69E0! \ge \sqrt {{{\left( {2x - 8 - 2x - 2} \right)}^2} + {{\left( {2y + 10 - 2y + 5} \right)}^2}} = 5\sqrt {13} \left( {\sqrt {{a^2} + {b^2}} + \sqrt {{c^2} + {d^2}} \ge \sqrt {{{\left( {a + c} \right)}^2} + {{\left( {b + d} \right)}^2}} } \right)\)

Dấu bằng xảy ra khi và chỉ khi \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca % WG4bGaai4oaiaadMhaaiaawIcacaGLPaaacqGH9aqpdaqadaqaamaa % laaabaGaeyOeI0IaaGinaiabgUcaRiaaigdacaaIYaWaaOaaaeaaca % aIZaaaleqaaaGcbaGaaGymaiaaiodaaaGaai4oamaalaaabaGaaGym % aiaaiMdacqGHsislcaaIXaGaaGioamaakaaabaGaaG4maaWcbeaaaO % qaaiaaigdacaaIZaaaaaGaayjkaiaawMcaaaaa!4A44! \left( {x;y} \right) = \left( {\frac{{ - 4 + 12\sqrt 3 }}{{13}};\frac{{19 - 18\sqrt 3 }}{{13}}} \right)\).

Suy ra _{\(% MathType!MTEF!2!1!+-
% feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa
% aaleaaciGGTbGaaiyAaiaac6gaaeqaaOGaeyypa0JaaGynamaakaaa
% baGaaGymaiaaiodaaSqabaGccqGHshI3caWGHbGaeyypa0JaaGynai
% aacUdacaWGIbGaeyypa0JaaGymaiaaiodadaGdKaWcbaaabeGccaGL
% sgcacaaIYaGaamyyaiabgUcaRiaadkgacqGH9aqpcaaIYaGaaG4maa
% aa!4DD1!
{A_{\min }} = 5\sqrt {13} \Rightarrow a = 5;b = 132a + b = 23\)}

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