Understanding the Circle Equation in Complex Numbers

The given equation is in terms of a complex number \(z\) and its conjugate \(\bar{z}\). This form often represents a circle in the complex plane. The general equation of a circle in complex form is given by:

\(z \bar{z} + \alpha \bar{z} + \bar{\alpha} z + k = 0\)

Here, \(\alpha\) is a complex number, and \(k\) is a real constant. For this equation to represent a circle, it must satisfy \(|\alpha|^2 - k > 0\). If it does, the center of the circle is \(-\alpha\) and the radius is \(\sqrt{|\alpha|^2 - k}\).

Analyzing the Given Complex Equation

The given equation is:

\(z \bar{z} + (3 - i)z + (3 + i)\bar{z} + 1 = 0\)

Let's compare this equation with the general form \(z \bar{z} + \alpha \bar{z} + \bar{\alpha} z + k = 0\).

• The coefficient of \(z\) is \((3 - i)\). Comparing with the general form, we have \(\bar{\alpha} = 3 - i\).
• The coefficient of \(\bar{z}\) is \((3 + i)\). Comparing with the general form, we have \(\alpha = 3 + i\). Note that \((3 + i)\) is indeed the conjugate of \((3 - i)\), so this is consistent.
• The constant term is \(1\). Comparing with the general form, we have \(k = 1\).

Determining the Center of the Circle

The center of the circle is given by \(-\alpha\). We found that \(\alpha = 3 + i\).

Center = \(-\alpha = -(3 + i) = -3 - i\).

In the complex plane, the complex number \(x + iy\) corresponds to the point \((x, y)\) in the Cartesian coordinate system. Therefore, the complex number \(-3 - i\) corresponds to the point \((-3, -1)\) in the Cartesian plane. This is the center of the circle in Cartesian coordinates.

Calculating the Radius of the Circle

The radius of the circle is given by the formula \(r = \sqrt{|\alpha|^2 - k}\).

We have \(\alpha = 3 + i\) and \(k = 1\).

First, let's calculate \(|\alpha|^2\):

\(|\alpha|^2 = |3 + i|^2\)

Recall that for a complex number \(a + bi\), the magnitude squared is \(|a + bi|^2 = a^2 + b^2\).

\(|3 + i|^2 = 3^2 + 1^2 = 9 + 1 = 10\)

Now, substitute the values of \(|\alpha|^2\) and \(k\) into the radius formula:

\(r = \sqrt{|\alpha|^2 - k} = \sqrt{10 - 1} = \sqrt{9}\)

\(r = 3\)

The radius of the circle is 3.

Summary of Circle Properties

Based on our calculations:

• Center of the circle: \(-3 - i\) (corresponding to point \((-3, -1)\))
• Radius of the circle: \(3\)

Matching with the Options

Let's compare our findings with the given options:

• Option 1: centre (-3, -1) and radius 3
• Option 2: centre (-3, 1) and radius 3
• Option 3: centre (-3, -1) and radius 4
• Option 4: centre (-3, 1) and radius 4

Our calculated center \((-3, -1)\) and radius \(3\) match Option 1.

Revision Table: Complex Circle Equation

Additional Information: Deriving the Complex Circle Form

The general equation of a circle in the Cartesian coordinate system is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Let \(z = x + iy\). Then \(\bar{z} = x - iy\).

• \(x = \frac{z + \bar{z}}{2}\)
• \(y = \frac{z - \bar{z}}{2i}\)

Let the center be represented by the complex number \(c = h + ik\). Then \(h = \frac{c + \bar{c}}{2}\) and \(k = \frac{c - \bar{c}}{2i}\).

Substituting \(x\) and \(y\) into the Cartesian equation:

\((\frac{z + \bar{z}}{2} - h)^2 + (\frac{z - \bar{z}}{2i} - k)^2 = r^2\)

A more direct way is to use the distance definition. A circle is the locus of points \(z\) such that the distance from a fixed center \(c\) is constant \(r\).

\(|z - c| = r\)

Squaring both sides:

\(|z - c|^2 = r^2\)

Using the property \(|w|^2 = w \bar{w}\):

\((z - c)(\overline{z - c}) = r^2\)

\((z - c)(\bar{z} - \bar{c}) = r^2\)

\(z \bar{z} - z \bar{c} - c \bar{z} + c \bar{c} = r^2\)

\(z \bar{z} - \bar{c} z - c \bar{z} + |c|^2 - r^2 = 0\)

Comparing this with the general form \(z \bar{z} + \bar{\alpha} z + \alpha \bar{z} + k = 0\), we can identify:

• \(\bar{\alpha} = -\bar{c}\), which means \(\alpha = -c\). The center is indeed \(c = -\alpha\).
• \(k = |c|^2 - r^2\).

From \(k = |c|^2 - r^2\), we can rearrange to find the radius squared: \(r^2 = |c|^2 - k\). Substituting \(c = -\alpha\), we get \(r^2 = |-\alpha|^2 - k = |\alpha|^2 - k\). Thus, the radius is \(r = \sqrt{|\alpha|^2 - k}\), confirming the formula used in the solution.