# Common tangent rules

Problem 29 on the JEE mains inspired this note:

Things to know

•Pythagorean theorem

•Graphical transformations (only vertical and horizontal shifts)

This note will describe a quick and easy way to find the number of common tangents to two circles.

Given the following equations for the circles:

$$(y-a)^2+(x-b)^2=r^2$$

$$y^2+x^2=s^2$$

find the number of common tangents these circles will have.

You may wonder, "this isn't generalized, you assume one is at the origin". Let me explain.

What I did was move both circles $$p$$ units along the x-axis and $$q$$ units along the y-axis to make one at the origin. The number of tangents to the circles won't change if we perform a transformation other than a distortion.

Now for the tangent rules: how many common tangents will they have if...

I) $$a^2+b^2=(r+s)^2$$ then there are 3 common tangents.

II) $$a^2+b^2>(r+s)^2$$ then there are 4 common tangents.

Now for case 3

III) $$a^2+b^2<(r+s)^2$$ then we have three cases

NOTE: you must check this first, don't immediately check the rules below (Quick rule, if r=s in this case, then there are 2 tangents).

III-1) $$a^2+b^2=(r-s)^2$$ then there is 1 common tangent.

III-2) $$a^2+b^2>(r-s)^2$$ then there are 2 common tangents.

III-3) $$a^2+b^2<(r-s)^2$$ then there are no common tangents.

Note by Trevor Arashiro
3 years ago

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Can you tell me such way in finding common tangents to a circle &a ellipse ?

- 3 years ago

Comment deleted Apr 07, 2015

That's why I made it a sub set of case 3. Because for all III, the second half of that inequality must be true.

- 3 years ago