If no solution of `3siny+12sin^3x=a` lies on the line `y=3x ,` then `a in (-oo,-9)uu(9,oo)` a`a in [-9,9]` `a a in {-9,9}` `non eoft h e s e`

Options

A. `a in (-oo, -9) uu (9, oo)`
B. `a in [-9, 9]`
C. `a in {-9, 9}`
D. none of these

Explanation

To solve the problem, we need to analyze the equation \(3 \sin y + 12 \sin^3 x = a\) and determine the conditions under which there are no solutions for this equation when \(y = 3x\).

### Step-by-Step Solution:

1. **Substituting \(y = 3x\)**:
We start by substituting \(y\) with \(3x\) in the equation:
\[ 3 \sin(3x) + 12 \sin^3 x = a \]

2. **Using the Triple Angle Formula**:
We know that:
\[ \sin(3x) = 3 \sin x - 4 \sin^3 x \]
Substituting this into the equation gives:
\[ 3(3 \sin x - 4 \sin^3 x) + 12 \sin^3 x = a \]

3. **Simplifying the Equation**:
Expanding and simplifying:
\[ 9 \sin x - 12 \sin^3 x + 12 \sin^3 x = a \]
This simplifies to:
\[ 9 \sin x = a \]

4. **Finding the Range of \(\sin x\)**:
The sine function, \(\sin x\), has a range of \([-1, 1]\). Therefore:
\[ -1 \leq \sin x \leq 1 \]
Multiplying through by 9 gives:
\[ -9 \leq 9 \sin x \leq 9 \]
This means:
\[ -9 \leq a \leq 9 \]

5. **Condition for No Solutions**:
For the equation \(9 \sin x = a\) to have no solutions, \(a\) must lie outside the range \([-9, 9]\). Thus, the values of \(a\) must satisfy:
\[ a < -9 \quad \text{or} \quad a > 9 \]

6. **Conclusion**:
Therefore, the values of \(a\) for which there are no solutions to the equation are:
\[ a \in (-\infty, -9) \cup (9, \infty) \]

### Final Answer:
The correct option is:
- \(a \in (-\infty, -9) \cup (9, \infty)\)