4. (10 points) For what values of the parameter $a$ (in centimeters) does the geometric locus of points $(x, y)$ in the coordinate plane, satisfying the equation

$(|x-2 a|+|x+2 a|+|2 y-a|+|2 y+a|)$ cm $=a^{2}-7$ cm $^{2}$, contain a rhombus of maximum area?

Can the area $S$ of this rhombus be greater than 100 cm²? If it is possible, specify the lengths of the diagonals of such a rhombus; if not, justify your answer.
## Solution.

The original equation is even in variables $x$ and $y$ and parameter $a \Rightarrow$ the sought geometric locus (GML) is symmetric with respect to the $x$, $y$ axes, and we can assume $a>0$.

In the region defined by the system of inequalities

$\left\{\begin{array}{c}0 \leq x \leq 2 a \\ 0 \leq y \leq \frac{a}{2}\end{array}\right.$, the original equation is equivalent to $a^{2}-6 a-7=0, a_{1,2}=-1$ or 7.

Thus, the necessary conditions for the parameter $a= \pm 1 ; \pm 7$.

However, the pair $a= \pm 1$ when substituted into the original equations gives $x, y \in \emptyset$.

In the other three regions of the first quadrant, the GML cannot contain an arbitrary figure with area $S>0$.

For $a=7$, we obtain the general GML in the form of a rectangle with a size of 28 along the $O x$ axis and 7 along the $O y$ axis, into which a rhombus can be inscribed. The area of this rhombus with diagonals $d_{1}=28$ and $d_{2}=7$ along the coordinate axes is calculated by the formula:

$S=\frac{1}{2} d_{1} d_{2}=98 \mathrm{~cm}^{2}$.

If the larger diagonal of the rhombus $d_{1}=28$ is stretched and rotated along the diagonal of the rectangle to $d_{1}^{*}=7 \sqrt{17}$ (while the smaller diagonal $d_{2}=7$ will also rotate, but it does not need to be stretched!),

then the area of such a rhombus $S^{*}=\frac{1}{2} d_{1}^{*} d_{2}=\frac{49 \sqrt{17}}{2}>101 \mathrm{~cm}^{2}$.

Since the original equation is even in $a$, the answer will also be $a=-7$.

Answer: $a= \pm 7$ cm; may be $d=7 \sqrt{17}$ and 7.