4. Given a circle with radius $r$ and one of its diameters $A B$. A tangent to the circle is drawn at point $B$, and a line through point $A$ intersects the circle at point $C$ and the tangent at point $D$. A line parallel to the tangent $B D$ is drawn through $C$, and a line parallel to the diameter $A B$ is drawn through $D$. These two lines intersect at point $E$. Determine the geometric locus of point $E$ if the chord $A C$ rotates around $A$.
Solution. Let us choose a coordinate system such that the equation of the circle is $x^{2}+y^{2}=r^{2}$. We choose the diameter $A B$ to lie on the $O x$-axis (see diagram). Then the equation of the tangent is $x=r$, and the equation of the chord $A C$ is $y=k(x+r)$. The coordinates of point $C$ will be

$$
x_{C}=r \frac{1-k^{2}}{1+k^{2}}, \quad y_{C}=\frac{2 r k}{1+k^{2}}
$$

and the coordinates of point $D$ will be $x_{D}=r$, $y_{D}=2 r k$. The equation of the line $D E$ will be

![](https://cdn.mathpix.com/cropped/2024_06_05_12f9fb0299f2ef2e3407g-8.jpg?height=500&width=509&top_left_y=1668&top_left_x=959)

$$
y=2 r k
$$

and the equation of the line $C E$ will be

$$
x=r \frac{(1-k)^{2}}{1+k^{2}}
$$

If we eliminate the parameter $k$ from (1) and (2), we will obtain

$$
y^{2}=4 r^{2} \frac{r-x}{r+x},
$$

which is the equation of the desired geometric locus.