Đáp án:
\(10 < m < \dfrac{{\sqrt {8545} + 25}}{{6\sqrt 3 }}\)
Giải thích các bước giải:
\(\begin{array}{l}
\left\{ \begin{array}{l}
3x - 6y = 1\\
5x - my = 22
\end{array} \right.\\
\to \left\{ \begin{array}{l}
15x - 30y = 5\\
15x - 3my = 66
\end{array} \right.\\
\to \left\{ \begin{array}{l}
\left( { - 30 + 3m} \right)y = - 61\\
x = \dfrac{{22 + my}}{5}
\end{array} \right.\\
\to \left\{ \begin{array}{l}
y = \dfrac{{ - 61}}{{ - 30 + 3m}}\\
x = \dfrac{{22 + m.\dfrac{{ - 61}}{{ - 30 + 3m}}}}{5} = \dfrac{{ - 660 + 66m - 30m + 3{m^2} - 61m}}{{5\left( { - 30 + 3m} \right)}}
\end{array} \right.\\
\to \left\{ \begin{array}{l}
y = \dfrac{{ - 61}}{{ - 30 + 3m}}\\
x = \dfrac{{3{m^2} - 25m - 660}}{{5\left( { - 30 + 3m} \right)}}
\end{array} \right.\\
Do:x < 0;y < 0\\
\to \left\{ \begin{array}{l}
\dfrac{{ - 61}}{{ - 30 + 3m}} < 0\\
\dfrac{{3{m^2} - 25m - 660}}{{5\left( { - 30 + 3m} \right)}} < 0
\end{array} \right.\\
\to \left\{ \begin{array}{l}
- 30 + 3m > 0\\
3{m^2} - 25m - 660 < 0
\end{array} \right.\\
\to \left\{ \begin{array}{l}
m > 10\\
3{m^2} - 2.m\sqrt 3 .\dfrac{{25}}{{2\sqrt 3 }} + \dfrac{{625}}{{12}} - \dfrac{{8545}}{{12}} < 0
\end{array} \right.\\
\to \left\{ \begin{array}{l}
m > 10\\
{\left( {m\sqrt 3 - \dfrac{{25}}{{2\sqrt 3 }}} \right)^2} < \dfrac{{8545}}{{12}}
\end{array} \right.\\
\to \left\{ \begin{array}{l}
m > 10\\
\left| {m\sqrt 3 - \dfrac{{25}}{{2\sqrt 3 }}} \right| = \sqrt {\dfrac{{8545}}{{12}}}
\end{array} \right.\\
\to \left\{ \begin{array}{l}
m > 10\\
\left( { - \sqrt {\dfrac{{8545}}{{12}}} + \dfrac{{25}}{{2\sqrt 3 }}} \right):3 < m < \left( {\sqrt {\dfrac{{8545}}{{12}}} + \dfrac{{25}}{{2\sqrt 3 }}} \right):3
\end{array} \right.\\
\to KL:10 < m < \dfrac{{\sqrt {8545} + 25}}{{6\sqrt 3 }}
\end{array}\)