Let a, b,c are respectively the sines and p, q, r are respectively the consines of `alpha, alpha+(2pi)/(3) and alpha+(4pi)/(3)`, then :
Q. The value of `(ab+bc+ca)` is :

Options

A. 0
B. `-(3)/(4)`
C. `-(1)/(2)`
D. -1

Answer

To solve the problem, we need to find the value of \( ab + bc + ca \) where \( a, b, c \) are the sines of the angles \( \alpha, \alpha + \frac{2\pi}{3}, \alpha + \frac{4\pi}{3} \) respectively.

### Step 1: Define the values of \( a, b, c \)
Let:
- \( a = \sin(\alpha) \)
- \( b = \sin\left(\alpha + \frac{2\pi}{3}\right) \)
- \( c = \sin\left(\alpha + \frac{4\pi}{3}\right) \)

### Step 2: Use the sine addition formula
Using the sine addition formula, we can express \( b \) and \( c \):
- \( b = \sin\left(\alpha + \frac{2\pi}{3}\right) = \sin(\alpha)\cos\left(\frac{2\pi}{3}\right) + \cos(\alpha)\sin\left(\frac{2\pi}{3}\right) \)
- \( c = \sin\left(\alpha + \frac{4\pi}{3}\right) = \sin(\alpha)\cos\left(\frac{4\pi}{3}\right) + \cos(\alpha)\sin\left(\frac{4\pi}{3}\right) \)

### Step 3: Calculate the values of \( \cos\left(\frac{2\pi}{3}\right) \) and \( \sin\left(\frac{2\pi}{3}\right) \)
- \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \)
- \( \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \)
- \( \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} \)
- \( \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} \)

### Step 4: Substitute the values of \( b \) and \( c \)
Now substituting these values into \( b \) and \( c \):
- \( b = \sin(\alpha) \left(-\frac{1}{2}\right) + \cos(\alpha) \left(\frac{\sqrt{3}}{2}\right) = -\frac{1}{2} \sin(\alpha) + \frac{\sqrt{3}}{2} \cos(\alpha) \)
- \( c = \sin(\alpha) \left(-\frac{1}{2}\right) + \cos(\alpha) \left(-\frac{\sqrt{3}}{2}\right) = -\frac{1}{2} \sin(\alpha) - \frac{\sqrt{3}}{2} \cos(\alpha) \)

### Step 5: Calculate \( ab + bc + ca \)
Now we can calculate \( ab + bc + ca \):
\[ ab = \sin(\alpha) \left(-\frac{1}{2} \sin(\alpha) + \frac{\sqrt{3}}{2} \cos(\alpha)\right) \]
\[ bc = \left(-\frac{1}{2} \sin(\alpha) + \frac{\sqrt{3}}{2} \cos(\alpha)\right)\left(-\frac{1}{2} \sin(\alpha) - \frac{\sqrt{3}}{2} \cos(\alpha)\right) \]
\[ ca = \left(-\frac{1}{2} \sin(\alpha) - \frac{\sqrt{3}}{2} \cos(\alpha)\right) \sin(\alpha) \]

### Step 6: Simplify the expression
After substituting and simplifying the expressions, we can combine like terms.

### Final Calculation
After performing all calculations and simplifications, we find that:
\[ ab + bc + ca = -\frac{3}{4} \]

### Conclusion
Thus, the final answer is:
\[ \boxed{-\frac{3}{4}} \]