If `|(z-25)/(z-1)| = 5 ` , the value of `|z|` is -

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To solve the equation \( \left| \frac{z - 25}{z - 1} \right| = 5 \), we will follow these steps:

### Step 1: Rewrite \( z \)
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then, we can express the equation as:
\[ \left| \frac{(x + iy) - 25}{(x + iy) - 1} \right| = 5 \]

### Step 2: Simplify the expression
This can be rewritten as:
\[ \left| \frac{(x - 25) + iy}{(x - 1) + iy} \right| = 5 \]

### Step 3: Use the modulus property
Using the property of modulus, we have:
\[ \frac{\sqrt{(x - 25)^2 + y^2}}{\sqrt{(x - 1)^2 + y^2}} = 5 \]

### Step 4: Cross-multiply
Cross-multiplying gives:
\[ \sqrt{(x - 25)^2 + y^2} = 5 \sqrt{(x - 1)^2 + y^2} \]

### Step 5: Square both sides
Squaring both sides results in:
\[ (x - 25)^2 + y^2 = 25((x - 1)^2 + y^2) \]

### Step 6: Expand both sides
Expanding both sides, we get:
\[ (x^2 - 50x + 625 + y^2) = 25(x^2 - 2x + 1 + y^2) \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ x^2 - 50x + 625 + y^2 = 25x^2 - 50x + 25 + 25y^2 \]
\[ x^2 - 50x + 625 + y^2 - 25x^2 + 50x - 25 - 25y^2 = 0 \]

### Step 8: Combine like terms
Combining like terms results in:
\[ -24x^2 - 24y^2 + 600 = 0 \]

### Step 9: Simplify the equation
Dividing through by -24 gives:
\[ x^2 + y^2 = 25 \]

### Step 10: Find \( |z| \)
Since \( |z| = \sqrt{x^2 + y^2} \), we have:
\[ |z| = \sqrt{25} = 5 \]

### Final Answer
Thus, the value of \( |z| \) is \( 5 \).

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