Answer
The central spot remains bright.
Work Step by Step
If the Michelson interferometer produces a bright central spot, then the path length difference must be $~~n~\lambda~~$ for some integer $n$
Suppose the light wavelength is changed to $\frac{\lambda}{2}$
Note that the path length difference is $~~(2n)~(\frac{\lambda}{2}),~~$ where $2n$ is an integer since $n$ is an integer
Therefore, the central spot remains bright.
