Locus of tangency points of tangents issued from a fixed point to a certain set of circles

Question

Let us fix a point $A$ on a fixed line segment $P B$. What is the set of points $X$ that can be obtained by taking a circle $o $ passing through $A$ and $B$,drawing a line through $P$ tangent to $o$ and calling this tangency point $X$?

Answer 1

Let $k=\sqrt{PA.PB}$.

Result: The answer is the circle with center $P$ and radius $k$ (see figure).

Let us call "pencil$_{AB}$" the set of circles passing through $A$ and $B$.

The short demonstration of the result above relies on the concept of power $Pow(P,(C))$ of a point $P$ wrt to a circle $(C)$. If you don't know it, and as we are in the case of a point which is outside all circles of "pencil$_{AB}$", have a look at (https://en.wikipedia.org/wiki/Power_of_a_point), where it is shown that there are two ays to compute the power:

$$Pow(P,(C))=PA.PB=PX^2.$$

Thus, whatever the circle of "pencil$_{AB}$", the power of $M$ will remain the same, because the product $PA.PB$ is a constant.