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:

\(\begin{array}{l}
\overrightarrow {MF}  = \frac{1}{2}.\left( {\overrightarrow {MF}  + \overrightarrow {MF} } \right) = \frac{1}{2}\left( {\overrightarrow {MN}  + \overrightarrow {NF}  + \overrightarrow {MD}  + \overrightarrow {DF} } \right) = \frac{1}{2}\left( {\overrightarrow {MN}  + \overrightarrow {MD} } \right) = \frac{1}{2}\left( {\overrightarrow a  + \overrightarrow b } \right)\\
\overrightarrow {NE}  = \overrightarrow {NM}  + \overrightarrow {ME}  = \overrightarrow {NM}  + k.\overrightarrow {MP}  =  - \overrightarrow a  + k.\overrightarrow b \\
MF \bot NE \Leftrightarrow \frac{1}{2}\left( {\overrightarrow a  + \overrightarrow b } \right)\left( { - \overrightarrow a  + k\overrightarrow b } \right) = 0\\
 \Leftrightarrow \left( {\overrightarrow a  + \overrightarrow b } \right)\left( { - \overrightarrow a  + k\overrightarrow b } \right) = 0\\
 \Leftrightarrow  - {\overrightarrow a ^2} + \left( {k - 1} \right)\overrightarrow a .\overrightarrow b  + k.{\overrightarrow b ^2} = 0\\
 \Leftrightarrow  - 16 + \left( {k - 1} \right).16 + 64k = 0\\
 \Leftrightarrow k = \frac{2}{5}
\end{array}\)