The complex number z satisfies thc condition `|z-25/z|=24`. The maximum distance from the origin of co-ordinates to the points z is

Options

A. 25
B. 30
C. 32
D. None of these

Answer

To solve the problem, we need to find the maximum distance from the origin of the complex number \( z \) that satisfies the condition \( |z - \frac{25}{z}| = 24 \).

### Step-by-Step Solution:

1. **Start with the given condition**:
\[ |z - \frac{25}{z}| = 24 \]

2. **Rewrite the expression**:
We can express \( z \) as \( |z| \) and rewrite the absolute value:
\[ |z| - |\frac{25}{z}| \leq |z - \frac{25}{z}| \leq |z| + |\frac{25}{z}| \]
Here, \( |\frac{25}{z}| = \frac{25}{|z|} \).

3. **Set up the inequality**:
From the condition, we have:
\[ |z| - \frac{25}{|z|} \leq 24 \quad \text{and} \quad |z| + \frac{25}{|z|} \geq 24 \]

4. **Solve the left inequality**:
\[ |z| - \frac{25}{|z|} \leq 24 \]
Multiply through by \( |z| \) (assuming \( |z| > 0 \)):
\[ |z|^2 - 25 \leq 24|z| \]
Rearranging gives:
\[ |z|^2 - 24|z| - 25 \leq 0 \]

5. **Use the quadratic formula**:
Let \( x = |z| \). The quadratic equation is:
\[ x^2 - 24x - 25 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{24 \pm \sqrt{(-24)^2 - 4 \cdot 1 \cdot (-25)}}{2 \cdot 1} \]
\[ x = \frac{24 \pm \sqrt{576 + 100}}{2} \]
\[ x = \frac{24 \pm \sqrt{676}}{2} \]
\[ x = \frac{24 \pm 26}{2} \]

6. **Calculate the roots**:
- First root:
\[ x = \frac{50}{2} = 25 \]
- Second root:
\[ x = \frac{-2}{2} = -1 \quad (\text{not valid since magnitude can't be negative}) \]

7. **Determine the maximum distance**:
Since \( |z| \) must satisfy the inequality \( |z|^2 - 24|z| - 25 \leq 0 \), we find that:
\[ |z| \leq 25 \]

8. **Conclusion**:
The maximum distance from the origin to the point \( z \) is:
\[ \text{Maximum distance} = 25 \]

### Final Answer:
Hence, the required maximum distance is \( 25 \) units.

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