If `z_(1) = 3+4i` and `z_(2) = 2+i`, and z satisfy the equation `2(z+bar(z)) + 3(z-bar(z))i = 0`, Then for Minimum value of `|z-z_(1)| + |z-z_(2)|`, possible value of z can be

Options

A. `(15)/(7) + (10)/(7)i`
B. `(15)/(7)+(8)/(7)i`
C. `(15)/(7)+(9)/(7)i`
D. `(10)/(7)+(15)/(7)i`

Explanation

To solve the problem, we need to find the complex number \( z \) that satisfies the equation:

\[ 2(z + \bar{z}) + 3(z - \bar{z})i = 0 \]

and minimizes the expression \( |z - z_1| + |z - z_2| \), where \( z_1 = 3 + 4i \) and \( z_2 = 2 + i \).

### Step 1: Simplify the given equation

First, we rewrite the equation using the definitions of \( z \) and \( \bar{z} \):

Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. Then, \( \bar{z} = x - yi \).

Substituting \( z \) and \( \bar{z} \) into the equation gives:

\[ 2((x + yi) + (x - yi)) + 3((x + yi) - (x - yi))i = 0 \]

This simplifies to:

\[ 2(2x) + 3(2yi)i = 0 \]

\[ 4x + 6yi^2 = 0 \]

Since \( i^2 = -1 \), we have:

\[ 4x - 6y = 0 \]

### Step 2: Solve for \( y \)

Rearranging the equation gives:

\[ 4x = 6y \implies y = \frac{2}{3}x \]

### Step 3: Express \( z \) in terms of \( x \)

Now we can express \( z \) as:

\[ z = x + \frac{2}{3}xi = x\left(1 + \frac{2}{3}i\right) \]

### Step 4: Find the minimum value of \( |z - z_1| + |z - z_2| \)

We need to minimize:

\[ |z - z_1| + |z - z_2| \]

Substituting \( z = x + \frac{2}{3}xi \):

\[ |z - z_1| = |(x - 3) + \left(\frac{2}{3}x - 4\right)i| \]
\[ |z - z_2| = |(x - 2) + \left(\frac{2}{3}x - 1\right)i| \]

### Step 5: Geometric interpretation

The expression \( |z - z_1| + |z - z_2| \) represents the sum of distances from the point \( z \) to the points \( z_1 \) and \( z_2 \). The minimum occurs when \( z \) lies on the line segment connecting \( z_1 \) and \( z_2 \).

### Step 6: Find the equation of the line connecting \( z_1 \) and \( z_2 \)

The slope \( m \) of the line through points \( (3, 4) \) and \( (2, 1) \) is:

\[ m = \frac{1 - 4}{2 - 3} = \frac{-3}{-1} = 3 \]

The equation of the line in point-slope form is:

\[ y - 4 = 3(x - 3) \]

### Step 7: Find the intersection with \( y = \frac{2}{3}x \)

To find the intersection of the line \( y = 3x - 5 \) and \( y = \frac{2}{3}x \), we set:

\[ 3x - 5 = \frac{2}{3}x \]

Multiplying through by 3 to eliminate the fraction:

\[ 9x - 15 = 2x \]
\[ 7x = 15 \implies x = \frac{15}{7} \]

Substituting back to find \( y \):

\[ y = \frac{2}{3} \cdot \frac{15}{7} = \frac{10}{7} \]

### Final Result

Thus, the possible value of \( z \) that minimizes \( |z - z_1| + |z - z_2| \) is:

\[ z = \frac{15}{7} + \frac{10}{7}i \]