The \(R C\) circuits in Figure P10-62 represent the situation at the input to an oscilloscope. The parallel combination of \(R_{1}\) and \(C_{1}\) represents the probe used to connect the oscilloscope to a test point. The parallel combination of \(R_{2}\) and \(C_{2}\) represents the input impedance of the oscilloscope.

(a) Assuming zero initial conditions, transform the circuit into the \(s\) domain and find the relationship between the test-point voltage \(V_{\mathrm{S}}(s)\) and the voltage \(V_{\mathrm{O}}(s)\) at the oscilloscope's input.

(b) For \(R_{2}=10 \mathrm{M} \Omega\) and \(C_{2}=5 \mathrm{pF}\), determine the values of \(R_{1}\) and \(C_{1}\) that make the input voltage a scaled