If area of `DeltaABC (Delta)` and angle C are given and if c opposite to given angle is minimum, then

Options

A. `a = sqrt((2Delta)/(sinC))`
B. `b = sqrt((2Delta)/(sinC))`
C. `a = (4Delta)/(sinC)`
D. `b = (4Delta)/(sin^(2)C)`

Explanation

To solve the problem, we need to find the condition under which the side \( c \) opposite to angle \( C \) is minimized, given the area \( \Delta \) of triangle \( ABC \) and angle \( C \).

### Step-by-Step Solution:

1. **Understanding the relationship between area, sides, and angles**:
The area \( \Delta \) of triangle \( ABC \) can be expressed using the formula:
\[ \Delta = \frac{1}{2}ab \sin C \]
where \( a \) and \( b \) are the lengths of the sides adjacent to angle \( C \).

2. **Rearranging the area formula**:
From the area formula, we can express \( ab \) in terms of \( \Delta \) and \( \sin C \):
\[ ab = \frac{2\Delta}{\sin C} \]

3. **Using the Law of Cosines**:
According to the Law of Cosines, we have:
\[ c^2 = a^2 + b^2 - 2ab \cos C \]

4. **Substituting \( ab \) into the Law of Cosines**:
We can substitute \( ab = \frac{2\Delta}{\sin C} \) into the equation for \( c^2 \):
\[ c^2 = a^2 + b^2 - 2 \left(\frac{2\Delta}{\sin C}\right) \cos C \]

5. **Minimizing \( c \)**:
To minimize \( c \), we need to minimize \( c^2 \). We can analyze the expression:
\[ c^2 = a^2 + b^2 - \frac{4\Delta \cos C}{\sin C} \]
To minimize \( c^2 \), we can use the condition that \( a \) and \( b \) should be equal (i.e., \( a = b \)). This is because, for a fixed area and angle, the triangle is most compact (and thus has the smallest \( c \)) when it is isosceles.

6. **Setting \( a = b \)**:
Let \( a = b \). Then we can denote \( a \) as \( A \):
\[ c^2 = 2A^2 - \frac{4\Delta \cos C}{\sin C} \]

7. **Finding the value of \( A \)**:
Using the area formula again:
\[ \Delta = \frac{1}{2}A^2 \sin C \]
Rearranging gives:
\[ A^2 = \frac{2\Delta}{\sin C} \]

8. **Final expression for \( c^2 \)**:
Substituting \( A^2 \) back into the expression for \( c^2 \):
\[ c^2 = 2 \left(\frac{2\Delta}{\sin C}\right) - \frac{4\Delta \cos C}{\sin C} \]
Simplifying this gives:
\[ c^2 = \frac{4\Delta}{\sin C} (1 - \cos C) \]

### Conclusion:
Thus, the minimum value of \( c \) occurs when \( a = b \) and is given by:
\[ c = \sqrt{\frac{4\Delta}{\sin C} (1 - \cos C)} \]