Non Inverting Summing Amplifier – Ripple Gain

You have a small mistake in the KCL equation at \$V_b\$: the denominator on the LHS is \$R_1\$, not \$R\$. This means $$V_b = \frac{R_2}{R_1 + R_2}V_1 + \frac{R_1}{R_1 + R_2}V_2$$

To prove this is a non-inverting summing amplifier, you need the relationship between the inputs and output. You know that \$V_a = V_b\$ so substitute the RHS of the above equation (which is \$V_a\$) into your equation for \$V_3\$:

$$V_3 = \left(\frac{R_f}{R}+1\right)V_a = \left(\frac{R_f}{R}+1\right)\left(\frac{R_2}{R_1 + R_2}V_1 + \frac{R_1}{R_1 + R_2}V_2\right)$$

The output is therefore a sum of \$V_1\$ and \$V_2\$ (each scaled by \$R_1\$ and \$R_2\$), which is then amplified by a factor of \$R_f/R + 1\$. The overall gain is positive (there are no minus signs) so the amplifier is non-inverting rather than inverting.

It may be easier to see this if you assume \$R_1 = R_2\$. Then

$$\begin{align}V_3 &= \left(\frac{R_f}{R}+1\right)\left(\frac{R_1}{R_1 + R_1}V_1 + \frac{R_1}{R_1 + R_1}V_2\right) = \left(\frac{R_f}{R}+1\right)\left(\frac{1}{2}V_1 + \frac{1}{2}V_2\right)\\ &= \frac{1}{2}\left(\frac{R_f}{R}+1\right)(V_1 + V_2)\end{align}$$

Now it is clearly the sum of \$V_1\$ and \$V_2\$ amplified by the (positive) gain $$\frac{1}{2}\left(\frac{R_f}{R}+1\right)$$

The cutoff frequency is the frequency at which the gain has fallen by -3dB over the gain at high frequencies.

(for the purposes of the below discussion, we'll ignore the high pass network R and C since the effects of R1/C1 are being examined)

If the signal was applied to C1 (rather than C1 being grounded) R2 would not matter at all- gain would be -R2/R1 for high frequencies. When the reactance of the C1 equals the resistance of R1 in magnitude you have a reduction in voltage of 1/sqrt(2) which is -3dB. R2 will determine the gain, but not the cutoff frequency.

However it's a bit more complex in this case, because the op-amp is being fed from the non-inverting input, and the gain is actually 1 + R2/R1 at high frequencies, so the frequency at which the gain is reduced by 1/sqrt(2) will actually depend on R2.

In an extreme case, if you short R2, it becomes a voltage follower and then Fc is limited only by the amplifier!