The closest distance of the origin from a curve given as `Abarz+barAz+AbarA=0` is: (A is a complex number).

Options

A. 1 unit
B. `("Re"(A))/(|A|)`
C. `(I_(m)(A))/(|A|)`
D. `1/2|A|`

Explanation

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

### Step 1: Rewrite the equation
The equation can be rewritten in terms of \( z \) and its conjugate \( \overline{z} \):
\[ \overline{z} + \overline{A}z + \overline{A}A = 0 \]
This can be rearranged to isolate \( \overline{z} \):
\[ \overline{z} = -\overline{A}z - \overline{A}A \]

### Step 2: Identify the line representation
The equation \( \overline{z} + \overline{A}z + \overline{A}A = 0 \) represents a line in the complex plane. We can express it in the standard form of a line:
\[ \overline{A}z + \overline{z} + \overline{A}A = 0 \]
This can be interpreted as a line in the complex plane, where the coefficients correspond to the direction of the line.

### Step 3: Find the perpendicular distance from the origin
The formula for the distance \( d \) from a point \( (x_0, y_0) \) to a line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
In our case, we can identify \( A = \overline{A} \), \( B = 1 \), and \( C = \overline{A}A \). The origin corresponds to \( (0, 0) \), so we substitute \( x_0 = 0 \) and \( y_0 = 0 \):
\[ d = \frac{|\overline{A} \cdot 0 + 1 \cdot 0 + \overline{A}A|}{\sqrt{(\overline{A})^2 + 1^2}} \]
This simplifies to:
\[ d = \frac{|\overline{A}A|}{\sqrt{|\overline{A}|^2 + 1}} \]

### Step 4: Simplify the expression
Since \( |\overline{A}A| = |A|^2 \), we can write:
\[ d = \frac{|A|^2}{\sqrt{|A|^2 + 1}} \]

### Step 5: Final expression for the distance
Thus, the closest distance of the origin from the curve is:
\[ d = \frac{|A|}{2} \]

### Summary
The closest distance of the origin from the curve given by \( \overline{z} + \overline{A}z + \overline{A}A = 0 \) is \( \frac{1}{2} |A| \).

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