Question

To find the position of the center of circle respect to $\overrightarrow{XY}$.

Short answer

The center lies in the perpendicular bisector $XY$ and is equidistant from the points $X$ and $Y$.

Step-by-step solution

Step 1. Given information. 

Line $l$; points $X$and $Y$ .

A circle through $Y$and tangent to $l$at $X$ .

Given figure,

 Step 2. Concept   Used.

Observe the figure.

Draw $\overrightarrow{XY}$.

Construct a perpendicular bisector of $\overrightarrow{XY}$as shown below.

Note that the center lies in the perpendicular bisector $\overrightarrow{XY}$and is equidistant from the points $X$and $Y$.