The equations of two tangents of an ellipse are:

$$
4 x+5 y=25, \quad 9 x+20 y=75
$$

What is the equation of the ellipse if its axes coincide with the coordinate axes?
The given equations can also be written as:

$$
\frac{4}{25} x+\frac{1}{5} y=1 \quad \text{and} \quad \frac{3}{25} x+\frac{4}{15} y=1
$$

If the equation of the sought ellipse is

$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
$$

and the coordinates of the points of tangency are \(x_{1}, y_{1}\) and \(x_{2}, y_{2}\), then the equations of the tangents are:

$$
\frac{x_{1}}{a^{2}} x+\frac{y_{1}}{b^{2}} y=1 \quad \text{and} \quad \frac{x_{2}}{a^{2}} x+\frac{y_{2}}{b^{2}} y=1
$$

By comparing these equations with the equations under (1), we see that:

$$
x_{1}=\frac{4}{25} a^{2}, \quad y_{1}=\frac{1}{5} b^{2}, \quad x_{2}=\frac{3}{25} a^{2}, \quad y_{2}=\frac{4}{15} b^{2}
$$

But the coordinates of the points of tangency also satisfy the equation of the ellipse, which is why:

$$
16 a^{2}+25 b^{2}=625
$$

and

$$
81 a^{2}+400 b^{2}=625 \cdot 9
$$

From these equations, \(a^{2}=25, b^{2}=9\) and thus the equation of the ellipse is:

$$
\frac{x^{2}}{25}+\frac{y^{2}}{9}=1
$$

(Júlia Perl, Györ.)

The problem was also solved by: Bayer B., Czank K., Filkorn J., Kerekes T., Krausz B., Gy. Krisztián, Krumpschink K., Gy. Lukhaub, Gy. Lupsa, G. Sasvári, J. Sasvári, T. Ulmer.