Analysis of two circles in circles

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Question

The number of common tangents to the circles $x^{2} + y^{2} - x = 0, x^{2} + y^{2} + x = 0$ , is

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Solution

The centres and radii of the circles are:
Centres: $C_{1} (1 / 2, 0) C_{2} (- 1 / 2, 0)$
Radii : $r_{1} = \frac{1}{2} r_{2} = \frac{1}{2}$
Clearly $C_{1} C_{2} = r_{1} + r_{2} .$
Therefore, the circles touch each other externally. Hence, there are 3 common tangents

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