If `a^3+b^3+6a b c=8c^3 & omega` is a cube root of unity then: (a)`a , b , c` are in `A.P.` (b) `a , b , c ,` are in `H.P.` (c) `a+bomega-2comega^2=0` (d) `a+bomega^2-2comega=0`

Options

A. `a,c,b` are in A.P
B. a,c,b are in H.P
C. `a+bomega - 2comega^(2) = 0`
D. `a+ bomega^(2) -2comega = 0`

Explanation

To solve the problem, we start with the equation given:

\[ a^3 + b^3 + 6abc = 8c^3 \]

### Step 1: Rearranging the Equation
We can rearrange the equation as follows:

\[ a^3 + b^3 - 8c^3 + 6abc = 0 \]

### Step 2: Using the Identity for Sums of Cubes
We can apply the identity for the sum of cubes, which states that:

\[ x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2 + y^2 + z^2 - xy - xz - yz) \]

In our case, let \( x = a \), \( y = b \), and \( z = -2c \). Thus, we can rewrite the equation:

\[ a^3 + b^3 + (-2c)^3 - 3ab(-2c) = 0 \]

This simplifies to:

\[ a^3 + b^3 - 8c^3 + 6abc = 0 \]

### Step 3: Setting Up the Condition
From the identity, we know that if \( a + b - 2c = 0 \), then:

\[ a + b = 2c \]

### Step 4: Concluding that \( a, b, c \) are in Arithmetic Progression (A.P.)
Since \( a + b = 2c \), we can conclude that \( a, b, c \) are in Arithmetic Progression (A.P.) because the average of \( a \) and \( b \) is \( c \).

### Step 5: Using the Properties of Cube Roots of Unity
Next, we consider the cube root of unity, denoted as \( \omega \), where:

\[ \omega^3 = 1 \quad \text{and} \quad 1 + \omega + \omega^2 = 0 \]

### Step 6: Applying the Cube Root of Unity
Using the property of cube roots of unity, we can write:

1. For \( a + b\omega - 2c\omega^2 = 0 \)
2. For \( a + b\omega^2 - 2c\omega = 0 \)

Both of these equations can be derived similarly to how we derived the A.P. condition.

### Final Conclusion
From the above steps, we conclude:

(a) \( a, b, c \) are in A.P. (Correct)

(c) \( a + b\omega - 2c\omega^2 = 0 \) (Correct)

(d) \( a + b\omega^2 - 2c\omega = 0 \) (Correct)

### Summary of Correct Options
The correct options are:
- (a) \( a, b, c \) are in A.P.
- (c) \( a + b\omega - 2c\omega^2 = 0 \)
- (d) \( a + b\omega^2 - 2c\omega = 0 \)

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