Let `alpha , beta , gamma ` parametric angles of 3 points P,Q and R respectively lying on `x^(2)+y^(2)=1 `. If the length of chords AP, PQ and AR are in GP where A is (-1,0), then `[Given , alpha , beta ,gamma in (0,2pi)]`.

Options

A. `sin""(alpha + gamma)/(4) cos""(alpha -gamma)/(4) ge sin""(beta)/(2)`
B. `sin((alpha + gamma)/(4)) cos((alpha -gamma)/(4)) le sin(beta)/(2)`
C. `sin""(alpha )/(2) sin""(gamma)/(2) ge sin""(beta)/(2)`
D. `sin""(alpha )/(2) sin""(gamma)/(2) le sin""(beta)/(2)`

Answer

To solve the problem, we need to find the lengths of the chords \( AP \), \( PQ \), and \( AR \) and show that they are in geometric progression (GP). The points \( P \), \( Q \), and \( R \) lie on the unit circle defined by the equation \( x^2 + y^2 = 1 \). The point \( A \) is given as \( (-1, 0) \).

### Step 1: Define the Points
Let:
- Point \( P \) be represented by the parametric coordinates \( P(\cos \alpha, \sin \alpha) \)
- Point \( Q \) be represented by the parametric coordinates \( Q(\cos \beta, \sin \beta) \)
- Point \( R \) be represented by the parametric coordinates \( R(\cos \gamma, \sin \gamma) \)

### Step 2: Calculate the Length of Chord \( AP \)
The length of chord \( AP \) can be calculated using the distance formula:
\[ AP = \sqrt{(\cos \alpha + 1)^2 + (\sin \alpha - 0)^2} \]
This simplifies to:
\[ AP = \sqrt{(\cos \alpha + 1)^2 + \sin^2 \alpha} \]
Expanding this, we have:
\[ AP = \sqrt{\cos^2 \alpha + 2\cos \alpha + 1 + \sin^2 \alpha} \]
Using the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \):
\[ AP = \sqrt{1 + 2\cos \alpha + 1} = \sqrt{2 + 2\cos \alpha} = \sqrt{2(1 + \cos \alpha)} = 2\cos\left(\frac{\alpha}{2}\right) \]

### Step 3: Calculate the Length of Chord \( AQ \)
Similarly, for chord \( AQ \):
\[ AQ = \sqrt{(\cos \beta + 1)^2 + (\sin \beta - 0)^2} \]
Following the same steps:
\[ AQ = \sqrt{2 + 2\cos \beta} = 2\cos\left(\frac{\beta}{2}\right) \]

### Step 4: Calculate the Length of Chord \( AR \)
For chord \( AR \):
\[ AR = \sqrt{(\cos \gamma + 1)^2 + (\sin \gamma - 0)^2} \]
Again, following the same steps:
\[ AR = \sqrt{2 + 2\cos \gamma} = 2\cos\left(\frac{\gamma}{2}\right) \]

### Step 5: Establish the Condition for GP
Since \( AP \), \( AQ \), and \( AR \) are in GP, we have:
\[ AQ^2 = AP \cdot AR \]
Substituting the values we found:
\[ (2\cos\left(\frac{\beta}{2}\right))^2 = (2\cos\left(\frac{\alpha}{2}\right))(2\cos\left(\frac{\gamma}{2}\right)) \]
This simplifies to:
\[ 4\cos^2\left(\frac{\beta}{2}\right) = 4\cos\left(\frac{\alpha}{2}\right)\cos\left(\frac{\gamma}{2}\right) \]
Dividing both sides by 4:
\[ \cos^2\left(\frac{\beta}{2}\right) = \cos\left(\frac{\alpha}{2}\right)\cos\left(\frac{\gamma}{2}\right) \]

### Final Condition
Thus, the condition for the lengths of the chords \( AP \), \( AQ \), and \( AR \) to be in GP is:
\[ \cos^2\left(\frac{\beta}{2}\right) = \cos\left(\frac{\alpha}{2}\right) \cos\left(\frac{\gamma}{2}\right) \]