If the equation `sin^2 x-a sin x + b = 0` has only one solution in `(0, pi)` then which of the following statements are correct?

Options

A. `a in (-oo, 1] uu [2, oo)`
B. `b in (-oo, 0] uu [1, oo)`
C. `a=1+b`
D. none of these

Explanation

To solve the problem, we need to analyze the quadratic equation given by:

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

We want to find the conditions under which this equation has only one solution in the interval \( (0, \pi) \).

### Step 1: Identify the nature of the roots
For a quadratic equation \( Ax^2 + Bx + C = 0 \) to have only one solution, the discriminant must be zero. The discriminant \( D \) is given by:

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

In our case, we have:
- \( A = 1 \)
- \( B = -a \)
- \( C = b \)

Thus, the discriminant becomes:

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

For the equation to have only one solution, we need:

\[ a^2 - 4b = 0 \]

### Step 2: Solve for \( b \)
From the equation \( a^2 - 4b = 0 \), we can express \( b \) in terms of \( a \):

\[ b = \frac{a^2}{4} \]

### Step 3: Analyze the range of \( \sin x \)
The function \( \sin x \) takes values in the interval \( [0, 1] \) for \( x \in (0, \pi) \). Therefore, for the quadratic equation to have real solutions, we need:

\[ 0 \leq \sin x \leq 1 \]

This implies that the roots of the equation \( \sin^2 x - a \sin x + b = 0 \) must lie within the interval \( [0, 1] \).

### Step 4: Determine conditions on \( a \) and \( b \)
Since we have \( b = \frac{a^2}{4} \), we need to ensure that both roots of the quadratic equation fall within the interval \( [0, 1] \).

1. The vertex of the parabola represented by the quadratic equation occurs at:

\[ \sin x = \frac{a}{2} \]

For the equation to have only one solution, the vertex must lie within the interval \( [0, 1] \):

\[ 0 \leq \frac{a}{2} \leq 1 \implies 0 \leq a \leq 2 \]

2. Additionally, since \( b = \frac{a^2}{4} \), we have:

\[ b \geq 0 \implies \frac{a^2}{4} \geq 0 \implies a \text{ can be any real number.} \]

### Conclusion
From the analysis, we conclude that:
- \( a \) must be in the range \( [0, 2] \).
- \( b \) must be non-negative and can be expressed as \( b = \frac{a^2}{4} \).