Read the following writeup carefully:
In argand plane `|z|` represent the distance of a point z from the origin. In general `|z_1-z_2|` represent the distance between two points `z_1 and z_2`. Also for a general moving point z in argand plane, if arg(z) `=theta`, then `z=|z|e^(itheta)`, where `e^(itheta) = cos theta + i sintheta`.
Now answer the following question
If `|z-(3+2i)|=|z cos ((pi)/(4) - "arg" z)|,` then locus of z is

Options

A. circle
B. parabola
C. ellipse
D. hyperbola

Answer

To solve the problem, we need to find the locus of the complex number \( z \) given the equation:

\[ |z - (3 + 2i)| = |z \cos\left(\frac{\pi}{4} - \arg z\right)| \]

### Step 1: Express \( z \) in terms of \( x \) and \( y \)

Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. The expression for the distance from the point \( (3, 2) \) in the Argand plane is:

\[ |z - (3 + 2i)| = |(x - 3) + i(y - 2)| = \sqrt{(x - 3)^2 + (y - 2)^2} \]

### Step 2: Express \( |z| \) and \( \arg z \)

The modulus of \( z \) is:

\[ |z| = \sqrt{x^2 + y^2} \]

The argument of \( z \) is given by:

\[ \arg z = \tan^{-1}\left(\frac{y}{x}\right) \]

### Step 3: Rewrite the right-hand side

The right-hand side of the equation involves \( \cos\left(\frac{\pi}{4} - \arg z\right) \). Using the cosine subtraction formula:

\[ \cos\left(\frac{\pi}{4} - \arg z\right) = \cos\left(\frac{\pi}{4}\right)\cos(\arg z) + \sin\left(\frac{\pi}{4}\right)\sin(\arg z) \]

Substituting \( \cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \):

\[ \cos\left(\frac{\pi}{4} - \arg z\right) = \frac{1}{\sqrt{2}}\left(\frac{x}{\sqrt{x^2 + y^2}} + \frac{y}{\sqrt{x^2 + y^2}}\right) = \frac{x + y}{\sqrt{2}\sqrt{x^2 + y^2}} \]

Thus, the right-hand side becomes:

\[ |z \cos\left(\frac{\pi}{4} - \arg z\right)| = |z| \cdot \cos\left(\frac{\pi}{4} - \arg z\right) = \sqrt{x^2 + y^2} \cdot \frac{x + y}{\sqrt{2}\sqrt{x^2 + y^2}} = \frac{x + y}{\sqrt{2}} \]

### Step 4: Set up the equation

Now we can equate both sides of the original equation:

\[ \sqrt{(x - 3)^2 + (y - 2)^2} = \frac{x + y}{\sqrt{2}} \]

### Step 5: Square both sides

Squaring both sides gives:

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

### Step 6: Expand and simplify

Expanding both sides:

\[ (x^2 - 6x + 9) + (y^2 - 4y + 4) = \frac{x^2 + 2xy + y^2}{2} \]

Combining terms:

\[ x^2 + y^2 - 6x - 4y + 13 = \frac{x^2 + 2xy + y^2}{2} \]

Multiply through by 2 to eliminate the fraction:

\[ 2x^2 + 2y^2 - 12x - 8y + 26 = x^2 + 2xy + y^2 \]

Rearranging gives:

\[ x^2 + y^2 - 2xy - 12x - 8y + 26 = 0 \]

### Step 7: Identify the locus

This is a conic section. To analyze it further, we can complete the square or use the general conic form.

### Final Result

The locus of \( z \) is a conic section, specifically a hyperbola.