I'm new to Adobe Illustrator and I wanted to accomplish this task. How do I remove the inner lines in my drawings? I tried with all the pathfinders effect but it keeps removing my tangent lines (the lines connecting both my circles).
It works like a charm as I followed the selected's answer guide:
Here it is.
Pathfinder won't find anything with just lines -- it only works with the area that a path is enclosing. You'll want to connect your tangent lines and make them into a solid shape, a rectangle in this case.
Use the white arrow tool to select the right-end anchors of both lines (click on the first, shift-click on the second) and then Object > Path > Join (or Ctrl/Opt + J). Do so again for the two right-end anchors to end up with a rectangle-like shape.
Select this shape and both circles and choose the first Pathfinder option 'Unite'.
There's a script for that, which has the added advantage of giving you perfect connecting tangent lines.
It's called Metaball by the awesome Sato Hiroyuki - you can download it as part of a big bundle of mostly useful Illustrator scripts from his page here.
It turns a selection of circles into singular blobs by following tangent lines. Here's an example from that page - you can also input a value for the curve, and if you input 100, it gives perfectly straight lines like in your example.
Because it creates closed path shapes, you can select the paths it creates along with the circles, use
Unite (the first of the Pathfinder window options), and it just works perfectly.