In a triangle ABC if `tan.(A)/(2)tan.(B)/(2)=(1)/(3)` and ab = 4, then the value of c can be

Options

A. 1
B. `1.5`
C. `2.5`
D. none of these

Explanation

To solve the problem, we start with the given equation and information:

**Given:**
1. \(\frac{\tan(A/2)}{\tan(B/2)} = \frac{1}{3}\)
2. \(ab = 4\)

We need to find the value of \(c\).

### Step 1: Use the tangent half-angle formula
We know that:
\[ \tan\left(\frac{A}{2}\right) = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} \]
\[ \tan\left(\frac{B}{2}\right) = \sqrt{\frac{(s-a)(s-c)}{s(s-b)}} \]
where \(s\) is the semi-perimeter of the triangle, given by \(s = \frac{a+b+c}{2}\).

### Step 2: Set up the equation
From the given ratio:
\[ \frac{\tan(A/2)}{\tan(B/2)} = \frac{\sqrt{(s-b)(s-c)}}{\sqrt{(s-a)(s-c)}} = \frac{1}{3} \]

Squaring both sides gives:
\[ \frac{(s-b)(s-c)}{(s-a)(s-c)} = \frac{1}{9} \]

### Step 3: Simplify the equation
Cross-multiplying gives:
\[ 9(s-b)(s-c) = (s-a)(s-c) \]

Expanding both sides:
\[ 9s^2 - 9bs - 9cs + 9bc = s^2 - as - cs + ac \]

### Step 4: Rearranging the terms
Rearranging gives:
\[ (9s^2 - s^2) + (as - 9bs) + (9cs - cs) + (9bc - ac) = 0 \]
\[ 8s^2 + (a - 9b)s + (9bc - ac) = 0 \]

### Step 5: Solve for \(s\)
Using the quadratic formula:
\[ s = \frac{-(a - 9b) \pm \sqrt{(a - 9b)^2 - 4 \cdot 8 \cdot (9bc - ac)}}{2 \cdot 8} \]

### Step 6: Substitute known values
Since \(ab = 4\), we can express \(b\) in terms of \(a\):
\[ b = \frac{4}{a} \]

Now substitute \(b\) into the equation and simplify to find \(c\).

### Step 7: Find \(c\)
After substituting and simplifying, we can find \(c\) in terms of \(a\).

### Conclusion
The value of \(c\) can be determined based on the values of \(a\) and \(b\) derived from the equations.