1. The line $x=a(a>0)$ and a circle with radius $a$ passing through the origin and tangent to the given line are given. A line is drawn through the origin, intersecting the circle at point $A(A \neq 0)$ and the given line at point $B$. Lines parallel to $O y$ and $O x$ are drawn through points $A$ and $B$, respectively. Let these lines intersect at point $M$. Find the geometric locus of points $M$.

![](https://cdn.mathpix.com/cropped/2024_06_05_a9dc3c06af03b935fe82g-6.jpg?height=805&width=591&top_left_y=1482&top_left_x=894)
The equation of the circle is $x^{2}+y^{2}-a x=0$, and the equation of an arbitrary line through the coordinate origin is $y=k x, k \in \mathbb{R}$. Therefore, we will have: $A\left(\frac{a}{1+k^{2}}, \frac{a k}{1+k^{2}}\right), B(a, a k)$.

For the coordinates $x, y$ of point $M$, we have: $x=\frac{2}{1+k^{2}}, y=a k$, from which we obtain (by eliminating $k$)

$$
x\left(a^{2}+y^{2}\right)=a^{3}
$$

which represents the equation of the desired geometric locus. The graph is given in the diagram.