If the multiplicative inverse of a complex number is ` (sqrt2 +5i)/17` ,then the complex number is

Options

A. `(sqrt2 -5i)/17`
B. `(sqrt2 +5i)/29`
C. `17/27 (sqrt2 -5i)`
D. `17/27 (sqrt2 +5i)`

Answer

To find the complex number given its multiplicative inverse, we can follow these steps:

### Step 1: Define the complex number
Let the complex number be \( z \).

### Step 2: Write the multiplicative inverse
The multiplicative inverse of \( z \) is given as:
\[ z^{-1} = \frac{\sqrt{2} + 5i}{17} \]

### Step 3: Express \( z \) in terms of its inverse
From the definition of multiplicative inverse, we know:
\[ z = \frac{1}{z^{-1}} \]
Substituting the value of \( z^{-1} \):
\[ z = \frac{1}{\frac{\sqrt{2} + 5i}{17}} = \frac{17}{\sqrt{2} + 5i} \]

### Step 4: Rationalize the denominator
To simplify \( z \), we multiply the numerator and the denominator by the conjugate of the denominator:
\[ z = \frac{17(\sqrt{2} - 5i)}{(\sqrt{2} + 5i)(\sqrt{2} - 5i)} \]

### Step 5: Calculate the denominator
Now, calculate the denominator using the formula \( (a + bi)(a - bi) = a^2 + b^2 \):
\[ (\sqrt{2})^2 + (5)^2 = 2 + 25 = 27 \]

### Step 6: Substitute back into the equation
Now substitute back into the equation for \( z \):
\[ z = \frac{17(\sqrt{2} - 5i)}{27} \]

### Step 7: Separate real and imaginary parts
Thus, we can express \( z \) as:
\[ z = \frac{17\sqrt{2}}{27} - \frac{85i}{27} \]

### Final Result
The complex number \( z \) is:
\[ z = \frac{17\sqrt{2}}{27} - \frac{85i}{27} \]

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