In a triangle `tan A+ tan B + tan C=6` and `tan A tan B= 2,` then the values of `tan A, tan B` and `tan C` are

Options

A. 1,2,3
B. `3,2//3,7//3`
C. `4,1//2,3//2`
D. none of these.

Explanation

To solve the problem, we need to find the values of \( \tan A \), \( \tan B \), and \( \tan C \) given that:

1. \( \tan A + \tan B + \tan C = 6 \)
2. \( \tan A \tan B = 2 \)

### Step-by-Step Solution:

**Step 1: Use the identity for the sum of tangents in a triangle.**

In a triangle, we have the identity:
\[ \tan A + \tan B + \tan C = \tan A \tan B \tan C \]
Given that \( \tan A + \tan B + \tan C = 6 \), we can express \( \tan C \) in terms of \( \tan A \) and \( \tan B \):
\[ \tan C = 6 - \tan A - \tan B \]

**Step 2: Substitute \( \tan C \) into the product identity.**

From the product identity, we know:
\[ \tan A \tan B = 2 \]
Now substituting \( \tan C \) into the identity:
\[ \tan A \tan B (6 - \tan A - \tan B) = 2 \]
Substituting \( \tan A \tan B = 2 \):
\[ 2(6 - \tan A - \tan B) = 2 \]
Dividing both sides by 2:
\[ 6 - \tan A - \tan B = 1 \]
Rearranging gives:
\[ \tan A + \tan B = 5 \]

**Step 3: Set up a system of equations.**

Now we have two equations:
1. \( \tan A + \tan B = 5 \)
2. \( \tan A \tan B = 2 \)

Let \( \tan A = x \) and \( \tan B = y \). Then we can rewrite the equations as:
1. \( x + y = 5 \)
2. \( xy = 2 \)

**Step 4: Solve the system of equations.**

From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 5 - x \]
Substituting this into the second equation:
\[ x(5 - x) = 2 \]
Expanding gives:
\[ 5x - x^2 = 2 \]
Rearranging gives:
\[ x^2 - 5x + 2 = 0 \]

**Step 5: Use the quadratic formula to find \( x \).**

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{5 \pm \sqrt{25 - 8}}{2} \]
\[ x = \frac{5 \pm \sqrt{17}}{2} \]

**Step 6: Find \( y \) values.**

Substituting back to find \( y \):
\[ y = 5 - x = 5 - \frac{5 \pm \sqrt{17}}{2} = \frac{5 \mp \sqrt{17}}{2} \]

**Step 7: Calculate \( \tan C \).**

Now we can find \( \tan C \):
\[ \tan C = 6 - (x + y) = 6 - 5 = 1 \]

**Final Values:**

Thus, the values are:
- \( \tan A = \frac{5 + \sqrt{17}}{2} \)
- \( \tan B = \frac{5 - \sqrt{17}}{2} \)
- \( \tan C = 1 \)