Answer
16 maxima will be detected.
Work Step by Step
At every point along the vertical line which bisects the two sources, the path length difference is zero, so the phase difference is zero. At all points on this vertical line, both above the x axis and below the x axis, there will be a maximum.
At all points on the horizontal line to the left of $S_1$ and to the right of $S_2$, the path length difference is $2.0~m$ so the phase difference is $4$ since $\lambda = 0.5~m$. At all of these points, there will be a maximum.
Similarly, there are three lines in each of the four quadrants such that the phase difference at all points along each line is 1, 2, and 3 respectively. At all points along these lines, there will be a maximum.
Any circle will intersect one point on each line. We can add all the points that any circle will meet:
$1+3+1+3+1+3+1+3 = 16$
16 maxima will be detected.
