Let's break down the problem step by step and solve it:

Part (a): Changing the 12-ohm resistor to 5 ohms

We are asked to find the equivalent resistance of two resistors in parallel when one resistor changes from 12 ohms to 5 ohms. In this case, the two resistors in parallel are:

1. $r_1 = 5$ ohms (instead of 12 ohms)
2. $r_2 = 4.8$ ohms (as determined from the previous analysis)

For resistors in parallel, the formula for the equivalent resistance $R_{\text{eq}}$ is:

$\frac{1}{R_{\text{eq}}} = \frac{1}{r_1} + \frac{1}{r_2}$

Substituting the values $r_1 = 5$ ohms and $r_2 = 4.8$ ohms:

$\frac{1}{R_{\text{eq}}} = \frac{1}{5} + \frac{1}{4.8}$

We can now calculate this:

$\frac{1}{R_{\text{eq}}} = 0.2 + 0.2083 \approx 0.4083$

So,

$R_{\text{eq}} \approx \frac{1}{0.4083} \approx 2.448 \, \text{ohms}$

Thus, the equivalent resistance when the 12-ohm resistor is replaced by a 5-ohm resistor is approximately 2.448 ohms.

Part (b): Both resistors are 8 ohms

Now, we calculate the equivalent resistance when both resistors are 8 ohms.

Again, using the formula for parallel resistors:

$\frac{1}{R_{\text{eq}}} = \frac{1}{r_1} + \frac{1}{r_2}$

Substituting $r_1 = 8$ ohms and $r_2 = 8$ ohms:

$\frac{1}{R_{\text{eq}}} = \frac{1}{8} + \frac{1}{8} = \frac{1}{4}$

Thus,

$R_{\text{eq}} = 4 \, \text{ohms}$

So, when both resistors are 8 ohms, the equivalent resistance is 4 ohms.

Part (c): Formula for the equivalent resistance of two resistors $r_1$ and $r_2$ in parallel

The general formula for the equivalent resistance $R_{\text{eq}}$ of two resistors $r_1$ and $r_2$ in parallel is:

$\frac{1}{R_{\text{eq}}} = \frac{1}{r_1} + \frac{1}{r_2}$

Solving for $R_{\text{eq}}$, we get:

$R_{\text{eq}} = \frac{r_1 r_2}{r_1 + r_2}$

Thus, the formula for the equivalent resistance of two resistors $r_1$ and $r_2$ in parallel is:

$R_{\text{eq}} = \frac{r_1 r_2}{r_1 + r_2}$

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Would you like further clarification or details on any of these steps? Also, here are some related questions to expand the understanding of parallel circuits:

1. What happens to the equivalent resistance if more resistors are added in parallel?
2. How do you calculate the total current if you know the total resistance in the circuit?
3. How can the formula for equivalent resistance be adapted for resistors in series?
4. What are the differences in behavior between resistors in series and resistors in parallel?
5. How can you measure the equivalent resistance in a physical circuit using a multimeter?

Tip: In parallel circuits, the equivalent resistance is always less than the smallest resistor in the parallel group.