The ratio in which a transverse common tangent drawn to two circles with radii 4 cm and 6 cm, respectively, divides the line joining their centres is:

2 : 3

Radius is perpendicular to tangent on circumference of circle. So,

∠P = ∠Q

By using alternate angle theorem ( Because OP is parallel to O'Q )

∠PCO = ∠QCO'

⇒ △PCO ≅ △PCO'

Hence,  \(\frac{PO}{QO'}\) = \(\frac{OC}{CO'}\)

We know, \(\frac{PO}{QO'}\) = \(\frac{4}{6}\)

⇒ \(\frac{OC}{CO'}\) = \(\frac{2}{3}\)

So, Required ratio in which transverse line divide the joining line = 2 : 3