A point is removed from the line $M N$ at a distance $a$. A circle with a given radius $r$ is described such that it passes through point $A$ and is tangent to the line $M N$. Find the distance between the resulting point of tangency and the given point $A$.

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Draw the diameter of the given circle through the obtained point of tangency and connect its ends with the given point A.

## Solution

Let $B$ be the obtained point of tangency, $C$ be the projection of point $A$ onto the line $M N$, $B D$ be the diameter of the given circle, and $K$ be the projection of point $A$ onto this diameter. Then $A K$ is the altitude of triangle $B A D$, drawn from the vertex of the right angle $D A B$. Therefore,

![](https://cdn.mathpix.com/cropped/2024_05_06_36c4956010bd91c4b4eeg-24.jpg?height=454&width=698&top_left_y=1008&top_left_x=680)

Answer

$\sqrt{2 a r}$.