If the equation `a sin^(3)x+(b-s)sin^(2)x+(c-b)sin x=c =0` has exactly three distinct solutions in `[0, pi]`, where a + b + c = 0, then which of the following is not the possible value of c/a ?

Options

A. 1
B. `(1)/(4)`
C. `(sqrt(2))/(7)`
D. `(2)/(7)`

Explanation

To solve the equation \( a \sin^3 x + (b - a) \sin^2 x + (c - b) \sin x - c = 0 \) for exactly three distinct solutions in the interval \([0, \pi]\), given that \( a + b + c = 0 \), we can follow these steps:

### Step 1: Rewrite the Equation
We start with the equation:
\[ a \sin^3 x + (b - a) \sin^2 x + (c - b) \sin x - c = 0 \]
We can factor out \( \sin x - 1 \):
\[ \sin x - 1 = 0 \quad \Rightarrow \quad \sin x = 1 \]
This gives us one solution, \( x = \frac{\pi}{2} \).

### Step 2: Factor the Polynomial
Now, we need to focus on the quadratic part:
\[ a \sin^2 x + b \sin x + c = 0 \]
Using the condition \( a + b + c = 0 \), we can express \( c \) in terms of \( a \) and \( b \):
\[ c = - (a + b) \]
Substituting this back into the quadratic equation gives:
\[ a \sin^2 x + b \sin x - (a + b) = 0 \]

### Step 3: Analyze the Quadratic Equation
The quadratic equation can be rewritten as:
\[ a \sin^2 x + b \sin x + (b + a) = 0 \]
For this quadratic to have two distinct solutions, the discriminant must be positive:
\[ D = b^2 - 4a(b + a) > 0 \]

### Step 4: Determine Conditions for Distinct Roots
Expanding the discriminant:
\[ D = b^2 - 4ab - 4a^2 > 0 \]
This implies that the quadratic must intersect the x-axis at two points, giving us a total of three distinct solutions (including \( \sin x = 1 \)).

### Step 5: Analyze the Values of \( \frac{c}{a} \)
From \( c = - (a + b) \), we can express \( \frac{c}{a} \):
\[ \frac{c}{a} = -\left(1 + \frac{b}{a}\right) \]
To find the possible values of \( \frac{c}{a} \), we need to analyze the conditions under which the discriminant is positive.

### Step 6: Evaluate Possible Values
Given the conditions for \( b \) in relation to \( a \):
1. If \( b = 0 \), then \( \frac{c}{a} = -1 \).
2. If \( b > 0 \), \( \frac{c}{a} < -1 \).
3. If \( b < 0 \), \( \frac{c}{a} > -1 \).

### Conclusion
The problem asks for which of the following is **not** a possible value of \( \frac{c}{a} \). The values provided are \( \frac{1}{4}, \frac{\sqrt{2}}{7}, \frac{2}{7}, 1 \).

Since \( \frac{c}{a} \) can be less than or equal to -1, the value \( 1 \) cannot be achieved. Therefore, the answer is:
\[ \text{The value of } \frac{c}{a} \text{ that is not possible is } 1. \]