Consider an ellipse with major and minor axes $2a$ and $2b$. From one of the vertices, we draw a chord, whose other endpoint is denoted by $M$. The position of the chord is to be determined from the condition that the difference between the chord and its projection onto the axis from which the chord was drawn is equal to a given length $l$. For which values of $l$ is the problem solvable?
We know the position of the chord if we know the coordinates of point $M$. Let these be $x_{1}$ and $y_{1}$. The equations for determining them are

$$
\begin{gathered}
b^{2} x_{1}^{2}+a^{2} y_{1}^{2}=a^{2} b^{2} \\
x_{1}^{2}+\left(b-y_{1}\right)^{2}=h^{2} \\
h-\left(b-y_{1}\right)=l
\end{gathered}
$$

if the chord is drawn from one end of the minor axis.

These equations can also be written in the following form

$$
\begin{gathered}
b^{2} x_{1}^{2}+a^{2} y_{1}^{2}-a^{2} b^{2}=0 \\
x_{1}^{2}+y_{1}^{2}-2 b y_{1}+b^{2}-h^{2}=0 \\
h-\left(b-y_{1}\right)=l
\end{gathered}
$$

From the first equation,

$$
x_{1}^{2}=\frac{a^{2} b^{2}-a^{2} y_{1}^{2}}{b^{2}}
$$

Substituting this into the second equation, we get

$$
\left(a^{2}-b^{2}\right) y_{1}^{2}+2 b^{3} y_{1}-b^{2}\left(a^{2}+b^{2}-h^{2}\right)=0
$$

But from the third equation,

$$
h^{2}=l^{2}+2 l b+b^{2}-2(l+b) y_{1}+y_{1}^{2}
$$

Substituting this value into equation (4), we get

$$
a^{2} y_{1}^{2}-2 b^{2} l y_{1}-b^{2}\left(a^{2}-2 b l-l^{2}\right)=0
$$

To determine the possibility of the problem, we must note that the values of $y_{1}$ must be real and lie between $+b$ and $-b$.

The reality condition is:

$$
4 b^{2}\left[\left(b^{2}-a^{2}\right) l^{2}-2 a^{2} b l+a^{4}\right] \geqq 0
$$

from which

$$
1 \leqq \frac{a^{2}}{a+b}
$$

Since substituting $b$ and $(-b)$ into the left side of the equation under (5) results in the positive

$$
b^{2} l^{2} \quad \text { and } \quad 4 b^{3} l+b^{2} l^{2}
$$

values, the roots of the equation will lie between $+b$ and $(-b)$ if their semi-sum falls within this interval,

that is, if

$$
-b<\frac{b^{2} l}{a}<b
$$

or, if

$$
-\frac{a^{2}}{b}<l<\frac{a^{2}}{b}
$$

But this is also satisfied for the inequality under (6), so (6) expresses the necessary and sufficient condition for the possibility of the problem.

Elemér Meitner, Vol. VIII, p. t. Budapest.