Đáp án đúng: A
Phương pháp giải:
+) Giả sử \(M\left( {{x_0};\,\,{y_0}} \right) \in \left( H \right)\).
Giải chi tiết:Giả sử \(M\left( {{x_0};\,\,{y_0}} \right) \in \left( H \right) \Rightarrow {x_0}{y_0} = 1\)\( \Rightarrow {y_0} = \frac{1}{{{x_0}}}\)\( \Rightarrow M\left( {{x_0};\,\,\frac{1}{{{x_0}}}} \right)\)
Ta có: \(M\left( {{x_0};\,\,\frac{1}{{{x_0}}}} \right)\) và \(A\left( {\frac{5}{2};\,\,\frac{5}{2}} \right)\)
\(\begin{array}{l}M{A^2} = {\left( {{x_0} - \frac{5}{2}} \right)^2} + {\left( {\frac{1}{{{x_0}}} - \frac{5}{2}} \right)^2}\,\, = x_0^2 - 5{x_0} + \frac{{25}}{4} + \frac{1}{{x_0^2}} - \frac{5}{{{x_0}}} + \frac{{25}}{4}\\\,\,\,\,\,\,\,\,\,\,\, = \left( {x_0^2 + \frac{1}{{x_0^2}}} \right) - \left( {5{x_0} + \frac{5}{{{x_0}}}} \right) + \frac{{25}}{2}\\\,\,\,\,\,\,\,\,\,\,\, = \left( {x_0^2 + 2 \cdot {x_0} \cdot \frac{1}{{{x_0}}} + \frac{1}{{x_0^2}}} \right) - 5 \cdot \left( {{x_0} + \frac{1}{{{x_0}}}} \right) - 2 + \frac{{25}}{2}\\\,\,\,\,\,\,\,\,\,\,\, = {\left( {{x_0} + \frac{1}{{{x_0}}}} \right)^2} - 5 \cdot \left( {{x_0} + \frac{1}{{{x_0}}}} \right) + \frac{{21}}{2}\\\,\,\,\,\,\,\,\,\,\,\, = {\left( {{x_0} + \frac{1}{{{x_0}}}} \right)^2} - 2 \cdot \left( {{x_0} + \frac{1}{{{x_0}}}} \right) \cdot \frac{5}{2} + \frac{{25}}{4} + \frac{{17}}{4}\\\,\,\,\,\,\,\,\,\,\,\, = {\left( {{x_0} + \frac{1}{{{x_0}}} - \frac{5}{2}} \right)^2} + \frac{{17}}{4}.\end{array}\)
\( \Rightarrow M{A^2}\, = {\left( {{x_0} + \frac{1}{{{x_0}}} - \frac{5}{2}} \right)^2} + \frac{{17}}{4} \ge \frac{{17}}{4}\)
Dấu “\( = \)” xảy ra \( \Leftrightarrow {x_0} + \frac{1}{{{x_0}}} = \frac{5}{2} \Leftrightarrow 2x_0^2 + 2 - 5{x_0} = 0\)\( \Leftrightarrow 2x_0^2 - 5{x_0} + 2 = 0 \Leftrightarrow \left[ \begin{array}{l}{x_0} = 2\\{x_0} = \frac{1}{2}\end{array} \right.\)
+) Với \({x_0} = 2 \Rightarrow {y_0} = \frac{1}{2} \Rightarrow M\left( {2;\,\,\frac{1}{2}} \right)\)
+) Với \({x_0} = \frac{1}{2} \Rightarrow {y_0} = 2 \Rightarrow M\left( {\frac{1}{2};\,\,2} \right)\)
Vậy \(M\left( {2;\,\,\frac{1}{2}} \right)\) hoặc \(M\left( {\frac{1}{2};\,\,2} \right)\).
Chọn A.
