`(1 + 7 i)/( (2 - i)^(2)) i n ( r , theta)` form is

Options

A. `( sqrt""2 , pi // 4)`
B. `( sqrt""2, pi//2)`
C. `( sqrt"" 2, 3 pi // 4)`
D. none

Answer

To convert the complex number \((1 + 7i)/( (2 - i)^{2})\) into the form \(r, \theta\), we will follow these steps:

### Step 1: Simplify the Denominator
First, we need to calculate \((2 - i)^{2}\).

\[ (2 - i)^{2} = 2^{2} - 2 \cdot 2 \cdot i + i^{2} = 4 - 4i + (-1) = 3 - 4i \]

### Step 2: Rewrite the Expression
Now we can rewrite the expression as:

\[ \frac{1 + 7i}{3 - 4i} \]

### Step 3: Multiply by the Conjugate
To simplify the fraction, we multiply the numerator and denominator by the conjugate of the denominator, which is \(3 + 4i\):

\[ \frac{(1 + 7i)(3 + 4i)}{(3 - 4i)(3 + 4i)} \]

### Step 4: Calculate the Denominator
Now, calculate the denominator:

\[ (3 - 4i)(3 + 4i) = 3^{2} - (4i)^{2} = 9 - 16(-1) = 9 + 16 = 25 \]

### Step 5: Calculate the Numerator
Next, calculate the numerator:

\[ (1 + 7i)(3 + 4i) = 1 \cdot 3 + 1 \cdot 4i + 7i \cdot 3 + 7i \cdot 4i = 3 + 4i + 21i + 28i^{2} \]
\[ = 3 + 25i - 28 = -25 + 25i \]

### Step 6: Combine the Results
Now, we can combine the results:

\[ \frac{-25 + 25i}{25} = -1 + i \]

### Step 7: Convert to Polar Form
Now we need to convert \(-1 + i\) into polar form \(r, \theta\).

#### Step 7.1: Calculate \(r\)
The modulus \(r\) is given by:

\[ r = \sqrt{(-1)^{2} + (1)^{2}} = \sqrt{1 + 1} = \sqrt{2} \]

#### Step 7.2: Calculate \(\theta\)
The argument \(\theta\) is given by:

\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1) \]

Since the point \(-1 + i\) is in the second quadrant, we add \(\pi\):

\[ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]

### Final Result
Thus, the polar form of the complex number is:

\[ \left(\sqrt{2}, \frac{3\pi}{4}\right) \]