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When I started Blending, most of my renders looked like the image below. Even with a white background the plane that was essential for shadows was making an ugly black line across my render.

However, it is very simple to remove the plane but keep the shadow, like this:

First you need to select your plane (by right clicking) and go into the Shading menu(F5)
Also make sure that your are in the Materials section 
Click Add New, go into the ‘shaders’ window and click Only Shad.

Your material preview will look like this:

Now render your image and you'll be free of ugly lines.
If the line is still there make sure that your light source is set to ray tracing and that ray tracing is turned on in the render options (Both of these options are on by default when Blender starts)