Complex Equation to Circle Transformation

The question asks us to identify the curve represented by the given equation involving complex numbers. The equation is: \[z \bar{z} + (1 + i) z + (1 - i) \bar{z} = 0\]

Understanding Complex Numbers

To convert this complex equation into a standard Cartesian form (in terms of \(x\) and \(y\)), we use the fundamental definitions for a complex number \(z\):

• Let \(z = x + iy\), where \(x\) is the real part and \(y\) is the imaginary part.
• The conjugate of \(z\) is \(\bar{z} = x - iy\).
• The product of \(z\) and its conjugate \(\bar{z}\) is \(z \bar{z} = (x + iy)(x - iy) = x^2 - (iy)^2 = x^2 - i^2 y^2\). Since \(i^2 = -1\), this simplifies to \(x^2 + y^2\). This term \(z \bar{z}\) represents the square of the magnitude of \(z\), i.e., \(|z|^2\).

Step-by-Step Conversion to Cartesian Form

Let's substitute \(z = x + iy\) and \(\bar{z} = x - iy\) into the given equation:

\[(x + iy)(x - iy) + (1 + i)(x + iy) + (1 - i)(x - iy) = 0\]

Now, we expand each term:

1. First Term: \(z \bar{z} = (x + iy)(x - iy) = x^2 + y^2\)
2. Second Term: \((1 + i)(x + iy)\)
   • Distribute 1: \(1 \cdot x + 1 \cdot iy = x + iy\)
   • Distribute i: \(i \cdot x + i \cdot iy = ix + i^2 y = ix - y\)
   • Combine: \(x + iy + ix - y = (x - y) + i(x + y)\)
3. Third Term: \((1 - i)(x - iy)\)
   • Distribute 1: \(1 \cdot x + 1 \cdot (-iy) = x - iy\)
   • Distribute -i: \(-i \cdot x -i \cdot (-iy) = -ix + i^2 y = -ix - y\)
   • Combine: \(x - iy - ix - y = (x - y) - i(x + y)\)

Substitute these expanded terms back into the original equation:

\[(x^2 + y^2) + [(x - y) + i(x + y)] + [(x - y) - i(x + y)] = 0\]

Next, we group the real and imaginary parts:

\[(x^2 + y^2 + x - y + x - y) + i(x + y - (x + y)) = 0\]

Simplify the expression:

• Real Part: \(x^2 + y^2 + 2x - 2y\)
• Imaginary Part: \(i(0) = 0\)

So, the Cartesian equation of the curve is:

\[x^2 + y^2 + 2x - 2y = 0\]

Identifying the Curve and its Properties

The general equation of a circle in Cartesian coordinates is given by: \[x^2 + y^2 + 2gx + 2fy + c = 0\]

Comparing our derived equation \(x^2 + y^2 + 2x - 2y = 0\) with the general form, we can identify the coefficients:

• \(2g = 2 \implies g = 1\)
• \(2f = -2 \implies f = -1\)
• \(c = 0\)

For a circle, the center is \((-g, -f)\) and the radius is \(\sqrt{g^2 + f^2 - c}\).

• Center of the circle: \((-g, -f) = (-1, -(-1)) = (-1, 1)\)
• Radius of the circle: \(r = \sqrt{g^2 + f^2 - c} = \sqrt{(1)^2 + (-1)^2 - 0} = \sqrt{1 + 1 - 0} = \sqrt{2}\)

Therefore, the curve represented by the given complex equation is a circle with its center at \((-1, 1)\) and a radius of \(\sqrt{2}\).

Summary of Results

This matches the description given in option 1.