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If the equations of two non intersecting circles are given in the general form i.e. $$x^2 + y^2 + 2gx +2fy + c =0$$ and $$x^2 + y^2 + 2g'x +2f'y + c' =0$$ , then how can I find common tangents on these circles. I am not able to figure out an approach for this. Also, what would be the lengths of these tangents

Bhavya
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  • @abiessu It would be much appreciated if i can get information on all four. But it's alright if you just give me a standard approach for finding them – Bhavya Jan 11 '17 at 04:26
  • Maybe easier to work with the form $(x-h)^2+(y-k)^2=r^2$ for the circles. – coffeemath Jan 11 '17 at 04:50
  • @coffeemath ok. So can u please show me the approach for that. I am taking the general form because it is used in many questions, but that can be easily converted into this format. Thanks – Bhavya Jan 11 '17 at 04:58
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    A possible approach: find the centers and the radii of such circles. With such info, you may easily find the interior and the exterior homothetic centers. The tangents through that points are the four common tangents. – Jack D'Aurizio Jan 11 '17 at 12:08
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    As an alternative, take the equation of a generic line and impose that is is tangent to both circles. That is equivalent to stating that two discriminants vanish at the same time. – Jack D'Aurizio Jan 11 '17 at 12:09
  • have a look at (http://math.stackexchange.com/q/211538) – Jean Marie Jan 14 '17 at 00:55

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