As ever, we draw a diagram of the situation that we are looking to investigate, such as the following diagram: 

We assume that our boat starts at the point (c,0), and always aims for a landing point at the origin of coordinates.  The speed of the boat is measured relative to the water, and we assume that the boat always “points” towards the landing point at the origin, so the direction of the velocity of the boat is towards the origin. 

To make progress, we need to express the motion of the boat relative to the (stationary) coordinate axes.  To do this, we need the components of the sum of the water velocity and the boat velocity, as shown in the blue inset in the above diagram.  Hence, for the geometry of the situation, the velocity components of the boat relative to the x and y axes are:

The minus sign in the first equation arises because the boat is heading in the negative x direction (towards the origin).  Now, using simple geometry,

so that 

This equation is easily solved, subject to , subject to x = c when y = 0, and we find that 

which can be re-arranged to give 

This is the path of the boat relative to the coordinate axes.  The following diagram shows the path of the boat for k = 0.5, i.e. the speed of the boat is twice that of the water, and a start point of (1,0).  

Clearly, the boat cannot land if k > 1, since in this case y diverges as x approaches zero.  This makes sense, as there is no way the boat can dock if the speed of the boat is less than the speed at which the water flows.  (By the way, what happens if k = 1?)