How many solutions system of equations, `arg (z + 3 -2i) = - pi/4 and |z + 4 | - |z - 3i| = 5 ` has ?

Explanation

To solve the system of equations given by `arg(z + 3 - 2i) = -π/4` and `|z + 4| - |z - 3i| = 5`, we will analyze each equation step by step.

### Step 1: Analyze the first equation
The first equation is `arg(z + 3 - 2i) = -π/4`.

Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. Then we can rewrite the equation as:
\[ arg((x + 3) + (y - 2)i) = -\frac{\pi}{4} \]
This means that the vector from the point \((-3, 2)\) to the point \((x, y)\) makes an angle of \(-\frac{\pi}{4}\) with the positive x-axis.

The slope of the line represented by this argument is given by:
\[ \tan\left(-\frac{\pi}{4}\right) = -1 \]
Thus, the line can be expressed as:
\[ y - 2 = -1(x + 3) \]
Simplifying this gives:
\[ y = -x - 1 \]

### Step 2: Analyze the second equation
The second equation is `|z + 4| - |z - 3i| = 5`.

Substituting \( z = x + yi \):
\[ |z + 4| = |(x + 4) + yi| = \sqrt{(x + 4)^2 + y^2} \]
\[ |z - 3i| = |x + (y - 3)i| = \sqrt{x^2 + (y - 3)^2} \]
Thus, the equation becomes:
\[ \sqrt{(x + 4)^2 + y^2} - \sqrt{x^2 + (y - 3)^2} = 5 \]

### Step 3: Rearranging the second equation
Rearranging gives:
\[ \sqrt{(x + 4)^2 + y^2} = 5 + \sqrt{x^2 + (y - 3)^2} \]
Squaring both sides:
\[ (x + 4)^2 + y^2 = (5 + \sqrt{x^2 + (y - 3)^2})^2 \]
Expanding both sides leads to:
\[ (x + 4)^2 + y^2 = 25 + 10\sqrt{x^2 + (y - 3)^2} + x^2 + (y - 3)^2 \]
This results in a more complex equation that describes a geometric relationship.

### Step 4: Geometric interpretation
The first equation represents a line with a slope of -1, while the second equation represents a hyperbola or a curve depending on the values of \( x \) and \( y \).

### Step 5: Finding intersections
To find the number of solutions, we need to check if the line intersects the curve represented by the second equation.

From the analysis, we can see that the line \( y = -x - 1 \) and the curve from the second equation do not intersect. The line moves in a different direction compared to the curve.

### Conclusion
Since the two geometric figures do not intersect, the system of equations has no solutions.

### Final Answer
**The system of equations has 0 solutions.**

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