As ever, the first stage to solving this problem is to draw a diagram that depicts the situation, such as the diagram shown below.  The string itself is shown in red and lies along the line AP.

One end of the string is initially located at the point (a,0), the other at (0,0).  The end of the string at (0,0) is dragged up the y-axis.  The other end of the string follows a path similar to the curve shown in black in the diagram above. 

It can be seen from this figure that when the end of the string not being dragged is at point P – coordinates (x,y) – the string itself (shown in red in the diagram) runs along the tangent to the path at the point P.   

The equation of the path is obtained by noting that the tangent at point P is given by: 

This equation is easily solved, subject to x = a when y = 0, to give 

Note that the "log" here is the natural logarithm.  The curve described by this equation is called a tractrix.  An example, with a = 1, is shown in the following figure: