Solving Complex Number Equations: Finding the Value of an Expression

This problem involves solving two equations related to the magnitude of complex numbers and then evaluating a specific expression involving the complex number \({\rm{z}}\). The given equations are:

1. \(\left| {\frac{{{\rm{z}} - 4}}{{{\rm{z}} - 8}}} \right| = 1\)
2. \(\left| {\frac{{\rm{z}}}{{{\rm{z}} - 2}}} \right| = \frac{3}{2}\)

We need to find the value of \(\left| {\frac{{{\rm{z}} - 6}}{{{\rm{z}} + 6}}} \right|\).

Step 1: Analyze the First Complex Number Equation

The first equation is \(\left| {\frac{{{\rm{z}} - 4}}{{{\rm{z}} - 8}}} \right| = 1\). Using the property that \(|\frac{w_1}{w_2}| = \frac{|w_1|}{|w_2|}\), we can rewrite this as:

$$ \frac{{\left| {{\rm{z}} - 4} \right|}}{{\left| {{\rm{z}} - 8} \right|}} = 1 $$

This implies \({\left| {{\rm{z}} - 4} \right|} = {\left| {{\rm{z}} - 8} \right|}\). The term \({\left| {{\rm{z}} - a} \right|}\) represents the distance between the complex number \({\rm{z}}\) and the complex number \(a\) in the complex plane. So, \({\left| {{\rm{z}} - 4} \right|} = {\left| {{\rm{z}} - 8} \right|}\) means that the complex number \({\rm{z}}\) is equidistant from the points representing the complex numbers 4 and 8.

The locus of points equidistant from two fixed points is the perpendicular bisector of the line segment connecting the two points. In the complex plane, 4 and 8 are points on the real axis. The segment connecting them is on the real axis, from (4, 0) to (8, 0). The midpoint of this segment is \(\frac{4+8}{2} = 6\). The perpendicular bisector is a vertical line passing through this midpoint. Thus, the real part of \({\rm{z}}\) must be 6.

Let \({\rm{z}} = x + iy\), where \(x\) is the real part and \(y\) is the imaginary part. The equation \({\left| {{\rm{z}} - 4} \right|} = {\left| {{\rm{z}} - 8} \right|}\) becomes:

$$ \left| {(x + iy) - 4} \right| = \left| {(x + iy) - 8} \right| $$ $$ \left| {(x - 4) + iy} \right| = \left| {(x - 8) + iy} \right| $$

Using the definition of magnitude, \(|a + ib| = \sqrt{a^2 + b^2}\):

$$ \sqrt{(x - 4)^2 + y^2} = \sqrt{(x - 8)^2 + y^2} $$

Squaring both sides:

$$ (x - 4)^2 + y^2 = (x - 8)^2 + y^2 $$ $$ x^2 - 8x + 16 + y^2 = x^2 - 16x + 64 + y^2 $$

Subtract \(x^2 + y^2\) from both sides:

$$ -8x + 16 = -16x + 64 $$ $$ 16x - 8x = 64 - 16 $$ $$ 8x = 48 $$ $$ x = 6 $$

So, from the first equation, we know that the real part of \({\rm{z}}\) is 6. Let \({\rm{z}} = 6 + iy\).

Step 2: Analyze the Second Complex Number Equation

The second equation is \(\left| {\frac{{\rm{z}}}{{{\rm{z}} - 2}}} \right| = \frac{3}{2}\). Using the magnitude property again:

$$ \frac{{\left| {\rm{z}} \right|}}{{\left| {{\rm{z}} - 2} \right|}} = \frac{3}{2} $$

Cross-multiplying gives \(2{\left| {\rm{z}} \right|} = 3{\left| {{\rm{z}} - 2} \right|}\). We substitute \({\rm{z}} = 6 + iy\) into this equation:

$$ 2{\left| {6 + iy} \right|} = 3{\left| {(6 + iy) - 2} \right|} $$ $$ 2{\left| {6 + iy} \right|} = 3{\left| {4 + iy} \right|} $$

Using the definition of magnitude:

$$ 2 \sqrt{6^2 + y^2} = 3 \sqrt{4^2 + y^2} $$ $$ 2 \sqrt{36 + y^2} = 3 \sqrt{16 + y^2} $$

Squaring both sides:

$$ (2 \sqrt{36 + y^2})^2 = (3 \sqrt{16 + y^2})^2 $$ $$ 4 (36 + y^2) = 9 (16 + y^2) $$ $$ 144 + 4y^2 = 144 + 9y^2 $$

Subtracting 144 from both sides:

$$ 4y^2 = 9y^2 $$ $$ 9y^2 - 4y^2 = 0 $$ $$ 5y^2 = 0 $$

This equation implies \(y^2 = 0\), so \(y = 0\).

Step 3: Determine the Complex Number \({\rm{z}}\)

From Step 1, we found that the real part of \({\rm{z}}\) is \(x=6\). From Step 2, we found that the imaginary part of \({\rm{z}}\) is \(y=0\). Therefore, the complex number \({\rm{z}}\) is \(6 + i(0) = 6\).

Step 4: Evaluate the Expression \({\left| {\frac{{{\rm{z}} - 6}}{{{\rm{z}} + 6}}} \right|}\)

Now we need to find the value of the given expression using \({\rm{z}} = 6\):

$$ \left| {\frac{{{\rm{z}} - 6}}{{{\rm{z}} + 6}}} \right| $$

Substitute \({\rm{z}} = 6\):

$$ \left| {\frac{{6 - 6}}{{{6} + 6}}} \right| = \left| {\frac{{0}}{{{12}}}} \right| $$ $$ \left| {0} \right| = 0 $$

The value of the expression is 0.

Summary of the Solution Steps

We used the properties of magnitude of complex numbers to solve the given equations. The first equation led us to determine the real part of \({\rm{z}}\) by interpreting it geometrically as the perpendicular bisector of the segment connecting 4 and 8. The second equation helped us find the imaginary part of \({\rm{z}}\). Once \({\rm{z}}\) was found, we substituted its value into the expression we needed to evaluate.

Revision Table: Key Complex Number Concepts

Additional Information: Loci in the Complex Plane

Understanding loci of complex numbers is crucial for solving geometric problems involving complex numbers. Here are a few common examples:

• Circle: The equation \(|{\rm{z}} - a| = r\) represents a circle centered at the complex number \(a\) with radius \(r\). All points \({\rm{z}}\) on this circle are at a constant distance \(r\) from the center \(a\).
• Line (Perpendicular Bisector): As seen in this problem, the equation \(|{\rm{z}} - a| = |{\rm{z}} - b|\) represents the perpendicular bisector of the line segment joining the points \(a\) and \(b\).
• Line Segment: The set of points \({\rm{z}}\) on the line segment joining \(a\) and \(b\) can be represented as \({\rm{z}} = (1-t)a + tb\) for \(0 \le t \le 1\), where \(t\) is a real number.
• Ellipse: The equation \(|{\rm{z}} - a| + |{\rm{z}} - b| = k\) (where \(k > |a - b|\)) represents an ellipse with foci at \(a\) and \(b\). The sum of the distances from any point on the ellipse to the two foci is constant (\(k\)).
• Hyperbola: The equation \(||{\rm{z}} - a| - |{\rm{z}} - b|| = k\) (where \(0 < k < |a - b|\)) represents a hyperbola with foci at \(a\) and \(b\). The absolute difference of the distances from any point on the hyperbola to the two foci is constant (\(k\)).

In this specific problem, the first equation \(|{\rm{z}} - 4| = |{\rm{z}} - 8|\) geometrically represents a line, which we found to be the vertical line where the real part is 6. The second equation \(2|{\rm{z}}| = 3|{\rm{z}} - 2|\) can also be interpreted geometrically, but algebraically solving for the imaginary part after fixing the real part was simpler in this case. The intersection of the two loci gives the specific complex number \({\rm{z}}\).