Hai điểm A,B thuộc hai nhánh của đồ thị \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2 % da9iaaiodacqGHRaWkdaWcaaqaaiaaiEdaaeaacaWG4bGaeyOeI0Ia % aG4maaaaaaa!3D0F! y = 3 + \frac{7}{{x - 3}}\). Khi đó độ dài đoạn thẳng AB ngắn nhất bằng bao nhiêu?

Đồ thị hàm số \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2 % da9iaaiodacqGHRaWkdaWcaaqaaiaaiEdaaeaacaWG4bGaeyOeI0Ia % aG4maaaaaaa!3D0F! y = 3 + \frac{7}{{x - 3}}\) đối xứng qua điểm I(3;3).

Hai điểm A,B thuộc hai nhánh của đồ thị có độ dài ngắn nhất khi A và B là giao điểm của đồ thị và đường thẳng y=x.

Ta có .\(3 + \frac{7}{{x - 3}} = x\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HS9aae % WaaeaacaWG4bGaeyOeI0IaaG4maaGaayjkaiaawMcaamaaCaaaleqa % baGaaGOmaaaakiabg2da9iaaiEdacqGHuhY2caWG4bWaaWbaaSqabe % aacaaIYaaaaOGaeyOeI0IaaGOnaiaadIhacqGHRaWkcaaIYaGaeyyp % a0JaaGimaaaa!498D! \Leftrightarrow {\left( {x - 3} \right)^2} = 7 \Leftrightarrow {x^2} - 6x + 2 = 0\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HS9aam % qaaqaabeqaaiaadIhacqGH9aqpcaaIZaGaey4kaSYaaOaaaeaacaaI % 3aaaleqaaOGaeyO0H4TaamyEaiabg2da9iaaiodacqGHRaWkdaGcaa % qaaiaaiEdaaSqabaaakeaacaWG4bGaeyypa0JaaG4maiabgkHiTmaa % kaaabaGaaG4naaWcbeaakiabgkDiElaadMhacqGH9aqpcaaIZaGaey % OeI0YaaOaaaeaacaaI3aaaleqaaaaakiaawUfaaaaa!5043! \Leftrightarrow \left[ \begin{array}{l} x = 3 + \sqrt 7 \Rightarrow y = 3 + \sqrt 7 \\ x = 3 - \sqrt 7 \Rightarrow y = 3 - \sqrt 7 \end{array} \right.\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taam % yqamaabmaabaGaaG4maiabgUcaRmaakaaabaGaaG4naaWcbeaakiaa % cUdacaaIZaGaey4kaSYaaOaaaeaacaaI3aaaleqaaaGccaGLOaGaay % zkaaGaaiilaiaadkeadaqadaqaaiaaiodacqGHsisldaGcaaqaaiaa % iEdaaSqabaGccaGG7aGaaG4maiabgkHiTmaakaaabaGaaG4naaWcbe % aaaOGaayjkaiaawMcaaiabgkDiElaadgeacaWGcbGaeyypa0JaaGOm % amaakaaabaGaaGymaiaaisdaaSqabaaaaa!5087! \Rightarrow A\left( {3 + \sqrt 7 ;3 + \sqrt 7 } \right),B\left( {3 - \sqrt 7 ;3 - \sqrt 7 } \right) \Rightarrow AB = 2\sqrt {14} \)

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