The two resistors shown in the circuit are referred to as in parallel. The total resistance of the resistors is given by the formula StartFraction 1 Over Upper R Subscript Upper T EndFraction
1
RTequals=StartFraction 1 Over Upper R 1 EndFraction plus StartFraction 1 Over Upper R 2 EndFraction
1
R1+
1
R2. Complete parts ​(a)dash–​(c).
Upper R 1R1equals=4 plus 2 i4+2i ​ohms, Upper R 2R2equals=1 plus i1+i ohms

Given the formula for the total resistance of two resistors in parallel:

\[
\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}
\]

where
\[
R_1 = 4 + 2i \text{ ohms and } R_2 = 1 + i \text{ ohms}
\]

We need to find the total resistance, \( R_T \), by performing the following steps.

### Step 1: Find \( \frac{1}{R_1} \) and \( \frac{1}{R_2} \)

1. **For** \( R_1 = 4 + 2i \):

   The reciprocal \( \frac{1}{R_1} \) can be found by multiplying the numerator and denominator by the conjugate of \( R_1 \), which is \( 4 - 2i \):

   \[
   \frac{1}{R_1} = \frac{1}{4 + 2i} = \frac{1 \cdot (4 - 2i)}{(4 + 2i)(4 - 2i)}
   \]

   Simplifying the denominator:
   \[
   (4 + 2i)(4 - 2i) = 16 - (2i)^2 = 16 + 4 = 20
   \]

   Therefore,
   \[
   \frac{1}{R_1} = \frac{4 - 2i}{20} = \frac{4}{20} - \frac{2i}{20} = \frac{1}{5} - \frac{i}{10}
   \]

2. **For** \( R_2 = 1 + i \):

   Similarly, find \( \frac{1}{R_2} \) by multiplying the numerator and denominator by the conjugate of \( R_2 \), which is \( 1 - i \):

   \[
   \frac{1}{R_2} = \frac{1 \cdot (1 - i)}{(1 + i)(1 - i)}
   \]

   The denominator simplifies as follows:
   \[
   (1 + i)(1 - i) = 1 - i^2 = 1 + 1 = 2
   \]

   Thus,
   \[
   \frac{1}{R_2} = \frac{1 - i}{2} = \frac{1}{2} - \frac{i}{2}
   \]

### Step 2: Add \( \frac{1}{R_1} \) and \( \frac{1}{R_2} \)

Now, we can find \( \frac{1}{R_T} \) by adding \( \frac{1}{R_1} \) and \( \frac{1}{R_2} \):

\[
\frac{1}{R_T} = \frac{1}{5} - \frac{i}{10} + \frac{1}{2} - \frac{i}{2}
\]

Combine real and imaginary parts:

1. Real part: \( \frac{1}{5} + \frac{1}{2} = \frac{2 + 5}{10} = \frac{7}{10} \)
2. Imaginary part: \( -\frac{i}{10} - \frac{i}{2} = -\frac{5i + i}{10} = -\frac{6i}{10} = -\frac{3i}{5} \)

Thus,

\[
\frac{1}{R_T} = \frac{7}{10} - \frac{3i}{5}
\]

### Step 3: Find \( R_T \) by taking the reciprocal of \( \frac{1}{R_T} \)

Let \( \frac{1}{R_T} = \frac{7}{10} - \frac{3i}{5} \), then

\[
R_T = \frac{1}{\frac{7}{10} - \frac{3i}{5}}
\]

Multiply the numerator and denominator by the conjugate of the denominator, \( \frac{7}{10} + \frac{3i}{5} \):

\[
R_T = \frac{1 \cdot \left( \frac{7}{10} + \frac{3i}{5} \right)}{\left( \frac{7}{10} \right)^2 + \left( \frac{3}{5} \right)^2}
\]

Solving this will yield the total resistance \( R_T \) in terms of a complex number.

The two resistors shown in the circuit are referred to as in parallel. The total resistance of the resistors is given by the formula StartFraction 1 Over Upper R Subscript Upper T EndFraction equals StartFraction 1 Over Upper R 1 EndFraction plus StartFraction 1 Over Upper R 2 EndFraction. Complete parts ​(a)dash–​(c). Upper R1 equals 4 plus 2i ohms, Upper R2 equals 1 plus i ohms.

Find the total resistance of two parallel resistors with complex impedances (4 + 2i ohms and 1 + i ohms) using the parallel resistance formula. This solution covers complex numbers and electrical circuit concepts, suitable for advanced high school students.

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