Suppose three real numbers `a`, `b`, `c` are in `G.P.` Let `z=(a+ib)/(c-ib)`. Then

Options

A. `z=(ib)/(c )`
B. `z=(ia)/(b)`
C. `z=(ia)/(c )`
D. `z=0`

Explanation

To solve the problem step by step, we start with the given information that three real numbers \( a, b, c \) are in geometric progression (G.P.). We are also given the expression for \( z \):

\[ z = \frac{a + ib}{c - ib} \]

### Step 1: Understanding the G.P. Condition
Since \( a, b, c \) are in G.P., we can express this condition mathematically. The relationship for G.P. is given by:

\[ \frac{b}{a} = \frac{c}{b} \implies b^2 = ac \]

### Step 2: Rewrite \( z \)
We can rewrite \( z \) by factoring out \( b \) from both the numerator and the denominator:

\[ z = \frac{a + ib}{c - ib} = \frac{b \cdot \frac{a}{b} + i}{b \cdot \frac{c}{b} - i} \]

This simplifies to:

\[ z = \frac{\frac{a}{b} + i}{\frac{c}{b} - i} \]

### Step 3: Substituting the G.P. Ratios
Let \( r = \frac{b}{a} \) (the common ratio). Then we can express \( \frac{a}{b} \) as \( \frac{1}{r} \) and \( \frac{c}{b} \) as \( r \):

\[ z = \frac{\frac{1}{r} + i}{r - i} \]

### Step 4: Finding a Common Denominator
To simplify \( z \), we will multiply the numerator and the denominator by the conjugate of the denominator:

\[ z = \frac{\left(\frac{1}{r} + i\right)(r + i)}{(r - i)(r + i)} \]

Calculating the denominator:

\[ (r - i)(r + i) = r^2 + 1 \]

Calculating the numerator:

\[ \left(\frac{1}{r} + i\right)(r + i) = \frac{1}{r}r + \frac{1}{r}i + ir + i^2 = 1 + \left(\frac{1}{r} + r\right)i - 1 = \left(\frac{1}{r} + r\right)i \]

Thus, we have:

\[ z = \frac{\left(\frac{1}{r} + r\right)i}{r^2 + 1} \]

### Step 5: Simplifying Further
Since \( r = \frac{b}{a} \), we can express \( \frac{1}{r} + r \):

\[ \frac{1}{r} + r = \frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} \]

Substituting this back into our expression for \( z \):

\[ z = \frac{\left(\frac{a^2 + b^2}{ab}\right)i}{r^2 + 1} \]

### Step 6: Final Expression
We can express \( z \) in terms of \( b \) and \( c \):

\[ z = \frac{ib}{c} \]

### Conclusion
Thus, we find that \( z \) can be expressed as:

\[ z = \frac{ib}{c} \]