When any two sides and one of the opposite acute angle are given, under certain additional conditions two triangles are possible. The case when two triangles are possible is called the ambiguous case.
In fact when any two sides and the angle opposite to one of them are given either no triangle is posible or only one triangle is possible or two triangles are possible.
In the ambiguous case, let a,b and `angle A` are given and `c_(1), c_(2)` are two values of the third side c.
On the basis of above information, answer the following questions
The difference between two values of c is

Options

A. A. `2sqrt((a ^(2)-b^(2)))`
B. B. `sqrt((a ^(2)-b^(2)))`
C. C. `2sqrt((a^(2)-b^(2)sin ^(2)A))`
D. D. `sqrt((a^(2)-b^(2)sin ^(2)A))`

Answer

To solve the problem, we need to find the difference between the two possible values of the third side \( c \) in the ambiguous case of triangles, where two sides \( a \) and \( b \) and the angle \( A \) opposite side \( a \) are given.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two sides \( a \) and \( b \), and an angle \( A \) opposite to side \( a \). We need to find the difference between the two possible values of the third side \( c \) (denoted as \( c_1 \) and \( c_2 \)).

2. **Using the Cosine Rule**:
According to the cosine rule, we can express the relationship between the sides and the angle:
\[ c^2 = a^2 + b^2 - 2ab \cos A \]
Rearranging gives us:
\[ 2ab \cos A = a^2 + b^2 - c^2 \]

3. **Forming a Quadratic Equation**:
We can rearrange the cosine rule to form a quadratic equation in terms of \( c \):
\[ c^2 - 2b \cos A \cdot c + (b^2 - a^2) = 0 \]
Here, \( A = 1 \), \( B = -2b \cos A \), and \( C = b^2 - a^2 \).

4. **Finding the Roots**:
The roots of the quadratic equation \( c^2 + Bc + C = 0 \) can be found using the quadratic formula:
\[ c_{1,2} = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Substituting our values:
\[ c_{1,2} = \frac{2b \cos A \pm \sqrt{(2b \cos A)^2 - 4 \cdot 1 \cdot (b^2 - a^2)}}{2} \]

5. **Calculating the Discriminant**:
The discriminant \( D \) is:
\[ D = (2b \cos A)^2 - 4(b^2 - a^2) = 4b^2 \cos^2 A - 4(b^2 - a^2) \]
Simplifying gives:
\[ D = 4b^2 \cos^2 A - 4b^2 + 4a^2 = 4(a^2 - b^2 \sin^2 A) \]

6. **Finding the Difference**:
The difference between the two values \( c_1 \) and \( c_2 \) is given by:
\[ |c_1 - c_2| = \frac{\sqrt{D}}{A} = \sqrt{4(a^2 - b^2 \sin^2 A)} = 2\sqrt{a^2 - b^2 \sin^2 A} \]

7. **Final Result**:
Therefore, the difference between the two values of \( c \) is:
\[ |c_1 - c_2| = 2\sqrt{a^2 - b^2 \sin^2 A} \]

### Conclusion:
The difference between the two values of \( c \) is \( 2\sqrt{a^2 - b^2 \sin^2 A} \).