If `sin theta=x+a/x" for all "x in R - {0}` , then which one of the following is correct ?

Options

A. `a ge4`
B. ` age1/2`
C. `ale1/4`
D. ` ale1/2`

Explanation

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

\[ \sin \theta = x + \frac{a}{x} \quad \text{for all } x \in \mathbb{R} - \{0\} \]

### Step 1: Rearranging the equation
We can rearrange the equation by multiplying both sides by \(x\) (since \(x \neq 0\)) to eliminate the fraction:

\[ x \sin \theta = x^2 + a \]

### Step 2: Forming a quadratic equation
Rearranging this gives us a quadratic equation in \(x\):

\[ x^2 - x \sin \theta + a = 0 \]

### Step 3: Identifying coefficients
In this quadratic equation, we can identify the coefficients:
- \(A = 1\)
- \(B = -\sin \theta\)
- \(C = a\)

### Step 4: Using the discriminant
For the quadratic equation to have real roots, the discriminant must be non-negative. The discriminant \(D\) is given by:

\[ D = B^2 - 4AC \]

Substituting the coefficients:

\[ D = (-\sin \theta)^2 - 4(1)(a) = \sin^2 \theta - 4a \]

### Step 5: Setting the discriminant condition
To ensure that the quadratic has real roots, we require:

\[ \sin^2 \theta - 4a \geq 0 \]

### Step 6: Rearranging the inequality
Rearranging the above inequality gives:

\[ \sin^2 \theta \geq 4a \]

### Step 7: Finding the bounds for \(a\)
Since \(\sin^2 \theta\) can take values in the range \([0, 1]\), we can deduce that:

\[ 4a \leq 1 \implies a \leq \frac{1}{4} \]

### Conclusion
Thus, we conclude that:

\[ a \leq \frac{1}{4} \]