Consider two circles ${\mathrm{S}}_1\equiv {\mathrm{x}}^2+{\mathrm{y}}^2+8\mathrm{x}=0$and ${\mathrm{S}}_2\equiv {\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}=0$ . Let $∆$PQR be formed by the common tangents to circles S₁ and S₂. Then which of the follwoing hold(s) good?

We have ${\mathrm{S}}_1\equiv {\mathrm{x}}^2+{\mathrm{y}}^2+8\mathrm{x}=0$
with centre (–4, 0) and radius = 4

$S_2\equiv x^2+y^2-2x=0$
with centre (1, 0) and radius = 1
Hence circles S₁ and S2 touch each other externally
Also $\mathrm{P}\left(\frac{8}{3},0\right)$
(As P divides line joining their centres in ratio of their radii externally)
Let equation of their direct common tangent be
$\mathrm{y}=\mathrm{m}\left(\mathrm{x}-\frac{8}{3}\right)$
By applying condition of tangency, we get

 

slope = ${\mathrm{m}}_1=\frac{3}{4}$

slope = ${\mathrm{m}}_2=\frac{-3}{4}$