Đáp án:
\[k = \frac{2}{5}\]
Giải thích các bước giải:
Đặt \(\overrightarrow {MN} = \overrightarrow a ;\,\,\,\,\overrightarrow {MP} = \overrightarrow b \)
Theo giả thiết ta có:
\(\begin{array}{l}
\left| {\overrightarrow a } \right| = 4;\,\,\,\,\left| {\overrightarrow b } \right| = 8\\
\overrightarrow a .\overrightarrow b = \left| {\overrightarrow a } \right|.\left| {\overrightarrow b } \right|.cos\left( {\overrightarrow a ;\overrightarrow b } \right) = 4.8.\cos 60^\circ = 16
\end{array}\)
Ta có:
\(\begin{array}{l}
\overrightarrow {MF} = \frac{1}{2}.\left( {\overrightarrow {MF} + \overrightarrow {MF} } \right) = \frac{1}{2}\left( {\overrightarrow {MN} + \overrightarrow {NF} + \overrightarrow {MD} + \overrightarrow {DF} } \right) = \frac{1}{2}\left( {\overrightarrow {MN} + \overrightarrow {MD} } \right) = \frac{1}{2}\left( {\overrightarrow a + \overrightarrow b } \right)\\
\overrightarrow {NE} = \overrightarrow {NM} + \overrightarrow {ME} = \overrightarrow {NM} + k.\overrightarrow {MP} = - \overrightarrow a + k.\overrightarrow b \\
MF \bot NE \Leftrightarrow \frac{1}{2}\left( {\overrightarrow a + \overrightarrow b } \right)\left( { - \overrightarrow a + k\overrightarrow b } \right) = 0\\
\Leftrightarrow \left( {\overrightarrow a + \overrightarrow b } \right)\left( { - \overrightarrow a + k\overrightarrow b } \right) = 0\\
\Leftrightarrow - {\overrightarrow a ^2} + \left( {k - 1} \right)\overrightarrow a .\overrightarrow b + k.{\overrightarrow b ^2} = 0\\
\Leftrightarrow - 16 + \left( {k - 1} \right).16 + 64k = 0\\
\Leftrightarrow k = \frac{2}{5}
\end{array}\)
