If a,b and c arethe sides of a traiangle such that `b.c =lamda ^(2),` then the relation is `a, lamdaand A` is

Options

A. `c ge 2 lamda sin((C)/(2))`
B. `b ge 2 lamda sin ((A)/(2))`
C. `a ge 2 lamdasin ((A)/(2))`
D. None of these

Answer

To solve the problem, we need to establish the relationship between the sides \(a\), \(b\), \(c\), and the angle \(A\) given that \(bc = \lambda^2\). We will use the cosine rule and some trigonometric identities to derive the required relation.

### Step-by-step Solution:

1. **Start with the Cosine Rule**:
The cosine rule states that:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Given \(bc = \lambda^2\), we can substitute this into the equation.

2. **Substituting \(bc\)**:
Substitute \(bc\) into the cosine rule:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2\lambda^2} \]

3. **Use the Half-Angle Formula**:
The half-angle formula for cosine can be expressed as:
\[ \cos A = 1 - 2\sin^2\left(\frac{A}{2}\right) \]
Setting the two expressions for \(\cos A\) equal gives:
\[ 1 - 2\sin^2\left(\frac{A}{2}\right) = \frac{b^2 + c^2 - a^2}{2\lambda^2} \]

4. **Rearranging the Equation**:
Rearranging the equation gives:
\[ 2\lambda^2 - 2\lambda^2 \cdot 2\sin^2\left(\frac{A}{2}\right) = b^2 + c^2 - a^2 \]
This simplifies to:
\[ 2\lambda^2 - b^2 - c^2 + a^2 = 4\lambda^2 \sin^2\left(\frac{A}{2}\right) \]

5. **Expressing in Terms of \(A\)**:
We can express the relationship in terms of \(A\):
\[ a^2 - (b^2 + c^2 - 2\lambda^2) = 4\lambda^2 \sin^2\left(\frac{A}{2}\right) \]

6. **Final Relation**:
The final relation we derive is:
\[ a^2 = b^2 + c^2 - 2\lambda^2 + 4\lambda^2 \sin^2\left(\frac{A}{2}\right) \]

### Conclusion:
The relationship between \(a\), \(\lambda\), and \(A\) is established through the cosine rule and the half-angle formula. The derived equation shows how the sides and angle are interconnected given the condition \(bc = \lambda^2\).