## Task B-4.4.

The set of all points that satisfy the equation $x^{2}+y^{2}=8|x|+6|y|$ is a curve in the plane. Determine the area of the part of the plane bounded by this curve. How many points $(x, y)$ with integer coordinates lie inside this curve (not on the curve) for which $x y>0$?
## Solution.

Let's write the given equation in the form

$$
|x|^{2}+|y|^{2}-8|x|-6|y|=0 \Leftrightarrow(|x|-4)^{2}+(|y|-3)^{2}=25
$$

For $x \geqslant 0, y \geqslant 0$, we have the part of the circle $(x-4)^{2}+(y-3)^{2}=25$ that lies in the first quadrant. For $x \leqslant 0, y \geqslant 0$, we have the part of the circle $(x+4)^{2}+(y-3)^{2}=25$ that lies in the second quadrant. For $x \leqslant 0, y \leqslant 0$, we have the part of the circle $(x+4)^{2}+(y+3)^{2}=25$ that lies in the third quadrant. For $x \geqslant 0, y \leqslant 0$, we have the part of the circle $(x-4)^{2}+(y+3)^{2}=25$ that lies in the fourth quadrant.

Let's draw these curves.

![](https://cdn.mathpix.com/cropped/2024_05_30_d18d2bbdee068dba95dfg-20.jpg?height=870&width=1008&top_left_y=1067&top_left_x=521)

Notice that the parts of the drawn circles are actually semicircles. Therefore, the area of the region bounded by the given set of points is

$$
P=4 \cdot \frac{5^{2} \pi}{2}+\frac{12 \cdot 16}{2}=50 \pi+96
$$

We also need to calculate the number of integer points within the drawn region, specifically the part that lies in the first and third quadrants.

The circle in the first quadrant passes through the points $(1,7),(4,8),(7,7),(8,6)$, and $(9,3)$. Now we can easily count the points that are inside the desired region in the first quadrant. The ordinate $y=7$ has 5 points, the ordinate $y=6$ has 7 points, and each of the ordinates $y \in\{1,2,3,4,5\}$ has 8 points.

Finally, we have a total of $(5+7+5 \cdot 8) \cdot 2=104$ points with integer coordinates within the desired region.

![](https://cdn.mathpix.com/cropped/2024_05_30_d18d2bbdee068dba95dfg-21.jpg?height=877&width=1016&top_left_y=321&top_left_x=497)