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 determine the minimum value of \(|\alpha - \beta|\) given 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 in G.P.**:
Let:
- \(A = \sqrt{5} \sin x\)
- \(B = 10 \sin x\)
- \(C = 10(4 \sin^2 x + 1)\)

For these to be in G.P., the condition \(B^2 = AC\) must hold.

2. **Set up the equation**:
\[ (10 \sin x)^2 = (\sqrt{5} \sin x)(10(4 \sin^2 x + 1)) \]
Simplifying this gives:
\[ 100 \sin^2 x = \sqrt{5} \sin x (40 \sin^2 x + 10) \]

3. **Rearranging the equation**:
Dividing both sides by \(\sin x\) (assuming \(\sin x \neq 0\)):
\[ 100 \sin x = \sqrt{5}(40 \sin^2 x + 10) \]
Rearranging yields:
\[ 4 \sin^2 x - 2\sqrt{5} \sin x + 1 = 0 \]

4. **Using the quadratic formula**:
For the quadratic equation \(4 \sin^2 x - 2\sqrt{5} \sin x + 1 = 0\), we can apply the quadratic formula:
\[ \sin x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 4\), \(b = -2\sqrt{5}\), and \(c = 1\):
\[ \sin x = \frac{2\sqrt{5} \pm \sqrt{(-2\sqrt{5})^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4} \]
\[ = \frac{2\sqrt{5} \pm \sqrt{20 - 16}}{8} \]
\[ = \frac{2\sqrt{5} \pm 2}{8} \]
\[ = \frac{\sqrt{5} \pm 1}{4} \]

5. **Finding values of \(x\)**:
Thus, we have two values for \(\sin x\):
- \(\sin x = \frac{\sqrt{5} + 1}{4}\)
- \(\sin x = \frac{\sqrt{5} - 1}{4}\)

We need to find the corresponding angles \(x\) in the interval \((0, \pi)\).

6. **Calculating angles**:
Let:
- \(\alpha = \sin^{-1}\left(\frac{\sqrt{5} + 1}{4}\right)\)
- \(\beta = \sin^{-1}\left(\frac{\sqrt{5} - 1}{4}\right)\)

Since both values are in the first quadrant, we can also consider the second quadrant values:
- \(\pi - \alpha\)
- \(\pi - \beta\)

7. **Finding \(|\alpha - \beta|\)**:
The minimum difference will occur between the two angles in the first quadrant:
\[ |\alpha - \beta| = \left|\sin^{-1}\left(\frac{\sqrt{5} + 1}{4}\right) - \sin^{-1}\left(\frac{\sqrt{5} - 1}{4}\right)\right| \]

Using known values, we find that:
\[ \alpha \approx 54^\circ \quad \text{and} \quad \beta \approx 18^\circ \]
Therefore:
\[ |\alpha - \beta| = 54^\circ - 18^\circ = 36^\circ \]

8. **Final answer**:
Converting \(36^\circ\) to radians:
\[ 36^\circ = \frac{\pi}{5} \]
Thus, the minimum value of \(|\alpha - \beta|\) is:
\[ \boxed{\frac{\pi}{5}} \]