If the sum of squares of roots of equation `x^(2)-(sin alpha-2)x-(1+sin alpha)=0` is the least, then `alpha` is equal to

Options

A. `pi//4`
B. `pi//3`
C. `pi//2`
D. `pi//6`

Explanation

To solve the problem, we need to find the value of \( \alpha \) such that the sum of squares of the roots of the quadratic equation

\[ x^2 - (\sin \alpha - 2)x - (1 + \sin \alpha) = 0 \]

is minimized.

### Step 1: Identify the coefficients of the quadratic equation
The given equation can be rewritten in the standard form \( ax^2 + bx + c = 0 \):
- \( a = 1 \)
- \( b = -(\sin \alpha - 2) = 2 - \sin \alpha \)
- \( c = -(1 + \sin \alpha) \)

### Step 2: Use Vieta's formulas to find the sum and product of the roots
Let the roots of the equation be \( \alpha \) and \( \beta \).
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = 2 - \sin \alpha \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = -(1 + \sin \alpha) \)

### Step 3: Find the sum of squares of the roots
The sum of squares of the roots can be expressed using the formula:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \]
Substituting the values from Vieta's formulas:
\[ \alpha^2 + \beta^2 = (2 - \sin \alpha)^2 - 2(-1 - \sin \alpha) \]

### Step 4: Simplify the expression
Now, we simplify the expression:
\[ \alpha^2 + \beta^2 = (2 - \sin \alpha)^2 + 2 + 2\sin \alpha \]
Expanding \( (2 - \sin \alpha)^2 \):
\[ = 4 - 4\sin \alpha + \sin^2 \alpha + 2 + 2\sin \alpha \]
Combining like terms:
\[ = \sin^2 \alpha - 2\sin \alpha + 6 \]

### Step 5: Find the minimum value
To minimize \( \sin^2 \alpha - 2\sin \alpha + 6 \), we can complete the square:
\[ = (\sin \alpha - 1)^2 + 5 \]
The minimum value occurs when \( (\sin \alpha - 1)^2 = 0 \), which implies:
\[ \sin \alpha = 1 \]

### Step 6: Determine the value of \( \alpha \)
The sine function equals 1 at:
\[ \alpha = \frac{\pi}{2} + 2k\pi \quad \text{for } k \in \mathbb{Z} \]
However, the principal value we are interested in is:
\[ \alpha = \frac{\pi}{2} \]

### Conclusion
Thus, the value of \( \alpha \) for which the sum of squares of the roots is minimized is:
\[ \alpha = \frac{\pi}{2} \]