In the given figure, PT is a tangent to the circle centered at O. OC is perpendicular to chord AB. Prove that PA . $\displaystyle \mathrm{PB}=\mathrm{PC}^{2}-\mathrm{AC}^{2}$.

\[\begin{aligned} & \mathbf{P A} \cdot \mathbf{P B}=(\mathbf{P C}-\mathbf{A C}) \cdot(\mathbf{P C}+\mathbf{B C}) \\ & =(\mathbf{P C}-\mathbf{A C})(\mathbf{P C}+\mathbf{A C}) \quad[\mathbf{A C}=\mathbf{B C}] \\ & =\mathbf{P C}^{2}-\mathbf{A C}^{2} \end{aligned} \]