Let `alpha, beta` be two distinct values of x lying in `(0,pi)` for which `sqrt5 sin x, 10 sin x, 10 (4 sin ^(2) x+1)` are 3 consecutive terms of a G.P. Then minimum value of `|alpha - beta|=`

Options

A. `pi/10`
B. `pi/5`
C. `(2pi)/(5)`
D. `(3pi)/(5 )`

Answer

To solve the problem, we need to find the minimum value of \(|\alpha - \beta|\) where \(\alpha\) and \(\beta\) are distinct values of \(x\) in the interval \((0, \pi)\) such that \(\sqrt{5} \sin x\), \(10 \sin x\), and \(10(4 \sin^2 x + 1)\) are three consecutive terms of a geometric progression (G.P.).

### Step-by-Step Solution:

1. **Identify the Terms**:
Let:
- \(a = \sqrt{5} \sin x\)
- \(b = 10 \sin x\)
- \(c = 10(4 \sin^2 x + 1)\)

2. **Use the G.P. Condition**:
For \(a\), \(b\), and \(c\) to be in G.P., we have:
\[ b^2 = ac \]

3. **Substitute the Values**:
Substitute \(a\), \(b\), and \(c\) into the G.P. condition:
\[ (10 \sin x)^2 = \sqrt{5} \sin x \cdot 10(4 \sin^2 x + 1) \]

4. **Simplify the Equation**:
This leads to:
\[ 100 \sin^2 x = 10 \sqrt{5} \sin x (4 \sin^2 x + 1) \]
Dividing both sides by \(10 \sin x\) (assuming \(\sin x \neq 0\)):
\[ 10 \sin x = \sqrt{5} (4 \sin^2 x + 1) \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ 4 \sqrt{5} \sin^2 x - 10 \sin x + \sqrt{5} = 0 \]

6. **Quadratic Form**:
This is a quadratic equation in \(\sin x\):
\[ 4 \sqrt{5} y^2 - 10y + \sqrt{5} = 0 \]
where \(y = \sin x\).

7. **Using the Quadratic Formula**:
The roots can be found using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 4\sqrt{5}\), \(b = -10\), and \(c = \sqrt{5}\):
\[ y = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 4\sqrt{5} \cdot \sqrt{5}}}{2 \cdot 4\sqrt{5}} \]
Simplifying gives:
\[ y = \frac{10 \pm \sqrt{100 - 80}}{8\sqrt{5}} = \frac{10 \pm \sqrt{20}}{8\sqrt{5}} = \frac{10 \pm 2\sqrt{5}}{8\sqrt{5}} = \frac{5 \pm \sqrt{5}}{4\sqrt{5}} \]

8. **Finding the Values of \(\sin x\)**:
Thus, the two values of \(\sin x\) are:
\[ \sin x_1 = \frac{5 + \sqrt{5}}{4\sqrt{5}}, \quad \sin x_2 = \frac{5 - \sqrt{5}}{4\sqrt{5}} \]

9. **Finding the Angles**:
The angles corresponding to these sine values in the interval \((0, \pi)\) are:
\[ x_1 = \arcsin\left(\frac{5 + \sqrt{5}}{4\sqrt{5}}\right), \quad x_2 = \arcsin\left(\frac{5 - \sqrt{5}}{4\sqrt{5}}\right) \]

10. **Calculating \(|\alpha - \beta|\)**:
The difference is:
\[ |\alpha - \beta| = |x_1 - x_2| \]

11. **Finding Minimum Value**:
To find the minimum value, we can calculate:
\[ |x_1 - x_2| = \left|\arcsin\left(\frac{5 + \sqrt{5}}{4\sqrt{5}}\right) - \arcsin\left(\frac{5 - \sqrt{5}}{4\sqrt{5}}\right)\right| \]

12. **Final Result**:
After evaluating the angles, we find that the minimum value of \(|\alpha - \beta|\) is:
\[ \frac{\pi}{5} \]