If the non-zero numbers a, b, c are in A.P. and `tan^(-1)a, tan^(-1)b, tan^(-1)c` are also in A.P., then

Options

A. a = b = c
B. `b^(2) = ac`
C. `a^(2) = bc`
D. `c^(2) = ab`

Answer

To solve the problem, we need to establish the relationship between the numbers \(a\), \(b\), and \(c\) given that they are in Arithmetic Progression (A.P.) and that \(\tan^{-1} a\), \(\tan^{-1} b\), and \(\tan^{-1} c\) are also in A.P.

### Step-by-Step Solution:

1. **Understanding A.P. Condition**:
Since \(a\), \(b\), and \(c\) are in A.P., we can express this condition mathematically:
\[ 2b = a + c \tag{1} \]

2. **Using the A.P. Condition for Inverse Tangents**:
The condition that \(\tan^{-1} a\), \(\tan^{-1} b\), and \(\tan^{-1} c\) are in A.P. can also be expressed mathematically:
\[ 2 \tan^{-1} b = \tan^{-1} a + \tan^{-1} c \tag{2} \]

3. **Using the Tangent Addition Formula**:
We know that:
\[ \tan(\tan^{-1} x + \tan^{-1} y) = \frac{x + y}{1 - xy} \]
Applying this to equation (2):
\[ \tan(2 \tan^{-1} b) = \tan(\tan^{-1} a + \tan^{-1} c) \]
The left-hand side can be expressed using the double angle formula:
\[ \tan(2 \tan^{-1} b) = \frac{2b}{1 - b^2} \]
The right-hand side becomes:
\[ \tan(\tan^{-1} a + \tan^{-1} c) = \frac{a + c}{1 - ac} \]

4. **Setting the Two Expressions Equal**:
Now we set the two expressions equal to each other:
\[ \frac{2b}{1 - b^2} = \frac{a + c}{1 - ac} \tag{3} \]

5. **Substituting Equation (1) into Equation (3)**:
From equation (1), we know \(a + c = 2b\). Substituting this into equation (3) gives:
\[ \frac{2b}{1 - b^2} = \frac{2b}{1 - ac} \]

6. **Cross Multiplying**:
Cross multiplying yields:
\[ 2b(1 - ac) = 2b(1 - b^2) \]
Assuming \(b \neq 0\) (since \(a\), \(b\), and \(c\) are non-zero), we can divide both sides by \(2b\):
\[ 1 - ac = 1 - b^2 \]

7. **Simplifying the Equation**:
This simplifies to:
\[ ac = b^2 \tag{4} \]

8. **Conclusion**:
From equations (1) and (4), we have:
- \(2b = a + c\) (A.P. condition)
- \(b^2 = ac\) (Geometric Progression condition)

The only scenario where both conditions hold true is when \(a = b = c\).

### Final Result:
Thus, we conclude that:
\[ a = b = c \]