Show that two circles S1: x2 + y2 − 4x + 2y + 1 = 0 and S2: x2 + y2 + 2x − 2y + 1 = 0 touch externally and find the equation of their common tangent T at the point of contact.

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Question 9:

a) The equations of two circles $S_1$ and $S_2$ are given by $S_1: x^2 + y^2 - 4x + 2y + 1 = 0$ and $S_2: x^2 + y^2 + 2x - 2y + 1 = 0$. Show that $S_1$ and $S_2$ touch externally and obtain the equation of the common tangent T at the point of contact.

Step 1: Find the centers and radii of the circles.
The general equation of a circle is $x^2 + y^2