If `z= x+iy` satisfies the equation arg `(z-2)="arg" (2z+3i)`, then `3x-4y` is equal to

Options

A. 5
B. `-3`
C. 7
D. 6

Answer

To solve the problem, we need to analyze the given equation involving the complex number \( z = x + iy \). We are given that:

\[ \arg(z - 2) = \arg(2z + 3i) \]

### Step 1: Rewrite the expressions
First, we express \( z - 2 \) and \( 2z + 3i \) in terms of \( x \) and \( y \):

\[ z - 2 = (x - 2) + iy \]
\[ 2z + 3i = 2(x + iy) + 3i = 2x + 2iy + 3i = 2x + i(2y + 3) \]

### Step 2: Write the arguments
Now, we can write the arguments of both expressions:

\[ \arg(z - 2) = \arg((x - 2) + iy) = \tan^{-1}\left(\frac{y}{x - 2}\right) \]
\[ \arg(2z + 3i) = \arg(2x + i(2y + 3)) = \tan^{-1}\left(\frac{2y + 3}{2x}\right) \]

### Step 3: Set the arguments equal
Since the arguments are equal, we have:

\[ \tan^{-1}\left(\frac{y}{x - 2}\right) = \tan^{-1}\left(\frac{2y + 3}{2x}\right) \]

### Step 4: Eliminate the tangent inverse
Since the tangent function is one-to-one in the range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), we can set the arguments equal to each other:

\[ \frac{y}{x - 2} = \frac{2y + 3}{2x} \]

### Step 5: Cross-multiply
Cross-multiplying gives us:

\[ y \cdot 2x = (x - 2)(2y + 3) \]

### Step 6: Expand and simplify
Expanding both sides:

\[ 2xy = 2xy + 3x - 4y - 6 \]

### Step 7: Rearranging the equation
Now, we can simplify this equation by canceling \( 2xy \) from both sides:

\[ 0 = 3x - 4y - 6 \]

### Step 8: Solve for \( 3x - 4y \)
Rearranging gives us:

\[ 3x - 4y = 6 \]

### Final Step: Conclusion
Thus, the value of \( 3x - 4y \) is:

\[ \boxed{6} \]