The value of `alpha` in `[0,2pi]` so that `x^(2)+y^(2)+2sqrt(sin alpha )x+(cos alpha-1)=0` having intercept on x-axis always greater than 2, is/are

Options

A. `(pi//4, 3pi//2)`
B. `(pi//4, (3pi)//4`
C. `(pi//4, 5pi//4)`
D. `[0,pi]`

Answer

To solve the problem, we need to analyze the given equation of the form:

\[ x^2 + y^2 + 2\sqrt{\sin \alpha} x + (\cos \alpha - 1) = 0 \]

This can be compared to the general equation of a circle:

\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

where \( g = \sqrt{\sin \alpha} \), \( f = 0 \), and \( c = \cos \alpha - 1 \).

### Step 1: Find the x-intercepts of the circle
The x-intercepts of the circle can be found using the formula:

\[ x_{\text{intercepts}} = 2\sqrt{g^2 - c} \]

Substituting the values of \( g \) and \( c \):

\[ x_{\text{intercepts}} = 2\sqrt{(\sqrt{\sin \alpha})^2 - (\cos \alpha - 1)} \]
\[ = 2\sqrt{\sin \alpha - (\cos \alpha - 1)} \]
\[ = 2\sqrt{\sin \alpha + 1 - \cos \alpha} \]

### Step 2: Set the condition for the x-intercepts
According to the problem, the x-intercepts must always be greater than 2:

\[ 2\sqrt{\sin \alpha + 1 - \cos \alpha} > 2 \]

Dividing both sides by 2:

\[ \sqrt{\sin \alpha + 1 - \cos \alpha} > 1 \]

### Step 3: Square both sides
To eliminate the square root, we square both sides:

\[ \sin \alpha + 1 - \cos \alpha > 1 \]

### Step 4: Simplify the inequality
Subtracting 1 from both sides gives:

\[ \sin \alpha - \cos \alpha > 0 \]

### Step 5: Analyze the inequality
This inequality implies:

\[ \sin \alpha > \cos \alpha \]

### Step 6: Determine the range of \( \alpha \)
To find the values of \( \alpha \) in the interval \( [0, 2\pi] \) where \( \sin \alpha > \cos \alpha \):

1. The sine function is greater than the cosine function in the interval \( \left(\frac{\pi}{4}, \frac{3\pi}{4}\right) \).
2. In this interval, \( \sin \alpha \) starts to exceed \( \cos \alpha \) at \( \alpha = \frac{\pi}{4} \) and remains greater until \( \alpha = \frac{3\pi}{4} \).

### Conclusion
Thus, the values of \( \alpha \) for which the x-intercepts of the circle are always greater than 2 are:

\[ \alpha \in \left(\frac{\pi}{4}, \frac{3\pi}{4}\right) \]