<s>[INST] The point of contact of between two circles and common tangent at this point. A large circle and a small circle have equations $x^2+y^2+2x-4y-27=0 $ and $x^2+y^2-12x+10y+43=0$ respectively. 
a) Show that the two circles externally touch at a single point and find the point of contact.
b) Establish the equation of the common tangent at this point.

Answers:
a) Proof and $(3,-2)$
b) $y=x-5$
I need explanation provided please.
 [/INST] $\mathcal{C_1}: (x+1)^2 + (y-2)^2 = (4\sqrt{2})^2\Rightarrow C_1 = (-1,2), r_1 = 4\sqrt{2}$.
$\mathcal{C_2}: (x-6)^2 + (y+5)^2 = (3\sqrt{2})^2\Rightarrow C_2 = (6,-5), r_2 = 3\sqrt{2}$.
$\overline{C_1C_2} = \sqrt{7^2+7^2} = 7\sqrt{2} = r_1+r_2\Rightarrow \mathcal{C_1}, \mathcal{C_2} $ are touching each other as claimed.
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