Task 1. In the coordinate plane $x O y$, a line $4 x+3 y=n$, $n>0$ is given, which is at a distance of 12 from the coordinate origin. Determine the area of the triangle formed by the line with the coordinate axes.
Solution. The right triangle formed with the coordinate axes has legs $\overline{O A}=a=\frac{n}{4}$ and $\overline{O B}=b=\frac{n}{3}$ (see diagram). Therefore, the length of the hypotenuse is

$$
\overline{A B}=c=\sqrt{a^{2}+b^{2}}=\sqrt{\frac{n^{2}}{16}+\frac{n^{2}}{9}}=\frac{5 n}{12}
$$

The height drawn to the hypotenuse is

![](https://cdn.mathpix.com/cropped/2024_06_05_9d1696c7fd1a2cbf5bb5g-11.jpg?height=421&width=486&top_left_y=1573&top_left_x=978)
12 units long, from which we get

$$
P=\frac{\frac{5 n}{12} \cdot 12}{2}=\frac{5}{2} n
$$

On the other hand, $P=\frac{a b}{2}=\frac{n^{2}}{24}$. Therefore,

$$
\frac{n^{2}}{24}=\frac{5 n}{2}
$$

Thus, $n=60$ and $P=150$.