I'd like help on the following problem:
There are three circles $A,B,O$ in the plane. The radius of $A$ is $12$, the radius of $B$ is $8$ and they are both inside of $O$. The centre of $O$ is the midpoint of the centres of $A$ and $B$. If $p,q$ ($p>q$) are the lengths of the lines which are tangent to $A$ and $B$, as shown in the diagram, then what is the value of $p^2 - q^2$?
I know the answer is $384$, but I can't seem to find why.
Consider the following diagram:
The right triangle with the red dashed hypotenuse of length $r_O$ and the dashed leg containing the center of circle $O$, has solid leg $$ \frac p2=\sqrt{r_O^2-\left(\frac{r_A-r_B}2\right)^2}\tag1 $$ The right triangle with the green dashed hypotenuse of length $r_O$ and the dashed leg containing the center of circle $O$, has solid leg $$ \frac q2=\sqrt{r_O^2-\left(\frac{r_A+r_B}2\right)^2}\tag2 $$ Therefore, $$ \begin{align} p^2-q^2 &=(r_A+r_B)^2-(r_A-r_B)^2\tag{3a}\\ &=4r_Ar_B\tag{3b} \end{align} $$