What is the central equation of that ellipse which is tangent to the lines $x+2 y=27$ and $7 x+4 y=81$?
$\mathrm{Az}$

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

the equation of the tangent line at point $\xi, \eta$ of the ellipse is:

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

This tangent intersects the $X$ and $Y$ axes at points $M$ and $N$, which are $m$ and $n$ units away from the origin, respectively, so from (2):

$$
\frac{m \xi}{a^{2}}=1
$$

and

$$
\frac{n \eta}{b^{2}}=1
$$

or equivalently:

$$
\frac{\xi}{a}=\frac{a}{m}
$$

and

$$
\frac{\eta}{b}=\frac{b}{n}
$$

thus

$$
\frac{a^{2}}{m^{2}}+\frac{b^{2}}{n^{2}}=\frac{\xi^{2}}{a^{2}}+\frac{\eta^{2}}{b^{2}}=1
$$

since $\xi, \eta$ is a point on the ellipse (1). If now the equations of the given lines are:

$$
\begin{aligned}
& A_{1} x+B_{1} y=C_{1} \\
& A_{2} x+B_{2} y=C_{2}
\end{aligned}
$$

$$
m_{1}=\frac{C_{1}}{A_{1}} ; \quad n_{1}=\frac{C_{1}}{B_{1}}
$$

and

$$
m_{2}=\frac{C_{2}}{A_{2}} ; \quad n_{2}=\frac{C_{2}}{B_{2}}
$$

then the following two equations hold based on (3):

$$
A_{1}^{2} a^{2}+B_{1}^{2} b^{2}=C_{1}^{2}
$$

and

$$
A_{2}^{2} a^{2}+B_{2}^{2} b^{2}=C_{2}^{2}
$$

from which:

$$
a^{2}=\frac{C_{1}^{2} B_{2}^{2}-C_{2}^{2} B_{1}^{2}}{A_{1}^{2} B_{2}^{2}-A_{2}^{2} B_{1}^{2}}=\frac{\left(C_{1} B_{2}+C_{2} B_{1}\right)\left(C_{1} B_{2}-C_{2} B_{1}\right)}{\left(A_{1} B_{2}+A_{2} B_{1}\right)\left(A_{1} B_{2}-A_{2} B_{1}\right)}
$$

and

$$
b^{2}=\frac{\left(A_{1} C_{2}+A_{2} C_{1}\right)\left(A_{1} C_{2}-A_{2} C_{1}\right)}{\left(A_{1} B_{2}+A_{2} B_{1}\right)\left(A_{1} B_{2}-A_{2} B_{1}\right)}
$$

Thus, the central equation of the ellipse that touches the lines (4) and (5) is obtained by substituting the values of $a^{2}$ and $b^{2}$ found here into (1), resulting in the following equation:

$$
\begin{gathered}
\left(A_{1} C_{2}+A_{2} C_{1}\right)\left(A_{1} C_{2}-A_{2} C_{1}\right) x^{2}+\left(C_{1} B_{2}+C_{2} B_{1}\right)\left(C_{1} B_{2}-C_{2} B_{1}\right) y^{2}= \\
=\left(A_{1} B_{2}+A_{2} B_{1}\right)\left(A_{1} B_{2}-A_{2} B_{1}\right)
\end{gathered}
$$

In our case:

$$
A_{1}=1 ; B_{1}=2 ; C_{1}=27 \quad \text { and } \quad A_{2}=7 ; B_{2}=4 \quad \text { and } \quad C_{2}=81
$$

thus the sought equation is:

$$
162 x^{2}+81 y^{2}=13122
$$

(Riesz Marcell, Györ.)

The problem was also solved by: Ádámffy E., Bartók I., Dömény E., Dömény I., Enyedi B., Haar A., Kürti I., Liebner A., Neidenbach E., Pám M., Pivnyik I., Rássy P., Riesz K., Rosenberg J., Schwemmer I., Szücs A.