Find the length of PC

Question

Here PE is the tangent of the two circle.
PA = 12 ; CD/AB = 2
Find the length of PC [Source: BDMO] ]1

Answer 1

By power of a point, $PA \cdot PD = PE^2$ and $PB \cdot PC = PE^2$. Thus, $$ \frac{PB}{PA} = \frac{PD}{PC}$$ and since $PB = PA + AB$ and $PD = PC + CD$, this implies $$ 1 + \frac{AB}{PA} = 1 + \frac{CD}{PC}.$$ Subtracting 1 from both sides and solving for $PC$ yields $$ PC = \frac{CD \cdot PA}{AB} = 2 \cdot 12 = \boxed{24}.$$

Answer 2

It holds that $$ PE^2= \boxed{PA \cdot PD = PB \cdot PC}.$$

Taking advantage of the equality in the box, we have:

$\begin{array}[t]{l} 12\cdot (PC + CD) = (12+AB) \cdot PC\\ 12 PC +12 CD = 12 PC + AB \cdot PC\\ PC = \dfrac{12CD}{AB}=24 \end{array}$