The closest distance of origin from the curve given by `b bar(z) + bar(b) z + b bar(b) = 0` (b is also a complex number ) is

Options

A. 1 unit
B. `(Re(b))/(|b|)`
C. `(IM(b))/(|b|)`
D. `(1)/(2)|b|`

Answer

To find the closest distance of the origin from the curve given by the equation \( b \bar{z} + \bar{b} z + b \bar{b} = 0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Equation**:
The equation of the curve is given as:
\[ b \bar{z} + \bar{b} z + b \bar{b} = 0 \]
Here, \( z \) is a complex number, and \( \bar{z} \) is its conjugate.

2. **Convert to Standard Form**:
Rearranging the equation, we have:
\[ b \bar{z} + \bar{b} z = -b \bar{b} \]
This can be interpreted as a line in the complex plane.

3. **Identify the Perpendicular Distance**:
The formula for the distance \( d \) from a point \( (x_0, y_0) \) to a line given by \( Ax + By + C = 0 \) is:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
In our case, we need to express the equation in this form.

4. **Substitute the Origin**:
The origin corresponds to the point \( (0, 0) \). Thus, substituting \( x_0 = 0 \) and \( y_0 = 0 \) into the distance formula gives:
\[ d = \frac{|b \cdot 0 + \bar{b} \cdot 0 + b \bar{b}|}{\sqrt{|b|^2 + |\bar{b}|^2}} \]
Since \( |\bar{b}| = |b| \), we can simplify this to:
\[ d = \frac{|b \bar{b}|}{\sqrt{2 |b|^2}} = \frac{|b|^2}{\sqrt{2} |b|} = \frac{|b|}{\sqrt{2}} \]

5. **Final Result**:
Therefore, the closest distance of the origin from the curve is:
\[ d = \frac{|b|}{\sqrt{2}} \]