Find the complex number z satisfying the equation `|(z-12)/(z-8i)|= (5)/(3), |(z-4)/(z-8)|=1`

Answer

To solve the equation \( \left| \frac{z - 12}{z - 8i} \right| = \frac{5}{3} \) and \( \left| \frac{z - 4}{z - 8} \right| = 1 \), we can follow these steps:

### Step 1: Define the complex number
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers.

### Step 2: Rewrite the first equation
The first equation can be rewritten as:
\[ \left| \frac{z - 12}{z - 8i} \right| = \frac{5}{3} \]
This implies:
\[ 3 |z - 12| = 5 |z - 8i| \]

### Step 3: Calculate the magnitudes
Now, we calculate the magnitudes:
- \( |z - 12| = |(x + yi) - 12| = |(x - 12) + yi| = \sqrt{(x - 12)^2 + y^2} \)
- \( |z - 8i| = |(x + yi) - 8i| = |x + (y - 8)i| = \sqrt{x^2 + (y - 8)^2} \)

### Step 4: Substitute the magnitudes into the equation
Substituting these into our equation gives:
\[ 3 \sqrt{(x - 12)^2 + y^2} = 5 \sqrt{x^2 + (y - 8)^2} \]

### Step 5: Square both sides
Squaring both sides to eliminate the square roots:
\[ 9((x - 12)^2 + y^2) = 25(x^2 + (y - 8)^2) \]

### Step 6: Expand both sides
Expanding both sides:
\[ 9(x^2 - 24x + 144 + y^2) = 25(x^2 + y^2 - 16y + 64) \]
This simplifies to:
\[ 9x^2 - 216x + 1296 + 9y^2 = 25x^2 + 25y^2 - 400y + 1600 \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ -16x^2 - 16y^2 + 216x - 400y + 304 = 0 \]
Dividing through by -16:
\[ x^2 + y^2 - \frac{27}{2}x + 25y - 19 = 0 \]

### Step 8: Rewrite the second equation
Now, consider the second equation:
\[ \left| \frac{z - 4}{z - 8} \right| = 1 \]
This implies:
\[ |z - 4| = |z - 8| \]

### Step 9: Calculate the magnitudes for the second equation
- \( |z - 4| = |(x + yi) - 4| = |(x - 4) + yi| = \sqrt{(x - 4)^2 + y^2} \)
- \( |z - 8| = |(x + yi) - 8| = |(x - 8) + yi| = \sqrt{(x - 8)^2 + y^2} \)

### Step 10: Set the magnitudes equal
Setting these equal gives:
\[ \sqrt{(x - 4)^2 + y^2} = \sqrt{(x - 8)^2 + y^2} \]

### Step 11: Square both sides
Squaring both sides:
\[ (x - 4)^2 + y^2 = (x - 8)^2 + y^2 \]

### Step 12: Simplify the equation
Cancelling \( y^2 \) and simplifying:
\[ (x - 4)^2 = (x - 8)^2 \]
Expanding gives:
\[ x^2 - 8x + 16 = x^2 - 16x + 64 \]
Rearranging gives:
\[ 8x - 48 = 0 \implies x = 6 \]

### Step 13: Substitute \( x \) back into the first equation
Substituting \( x = 6 \) into the first equation:
\[ 6^2 + y^2 - \frac{27}{2}(6) + 25y - 19 = 0 \]
This simplifies to:
\[ 36 + y^2 - 81 + 25y - 19 = 0 \implies y^2 + 25y - 64 = 0 \]

### Step 14: Solve for \( y \)
Using the quadratic formula:
\[ y = \frac{-25 \pm \sqrt{25^2 - 4 \cdot 1 \cdot (-64)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ y = \frac{-25 \pm \sqrt{625 + 256}}{2} = \frac{-25 \pm \sqrt{881}}{2} \]

### Step 15: Find the values of \( z \)
Thus, we have two possible values for \( y \):
1. \( y_1 = \frac{-25 + \sqrt{881}}{2} \)
2. \( y_2 = \frac{-25 - \sqrt{881}}{2} \)

Finally, the complex numbers \( z \) are:
1. \( z_1 = 6 + \left(\frac{-25 + \sqrt{881}}{2}\right)i \)
2. \( z_2 = 6 + \left(\frac{-25 - \sqrt{881}}{2}\right)i \)