Transverse common tangents are drawn from O to the two circles `C_1,C_2` with 4, 2 respectively. Then the ratio of the areas of triangles formed by the tangents drawn from O to the circles `C_1 and C_2` and chord of contacts of O w.r.t the circles `C_1 and C_2` respectively is

Options

A. 3 units
B. 6 units
C. 4 units
D. 5 units

Explanation

To solve the problem, we need to find the ratio of the areas of triangles formed by the tangents drawn from point O to the circles \( C_1 \) and \( C_2 \) and the chord of contacts of O with respect to the circles \( C_1 \) and \( C_2 \) respectively.

### Step-by-Step Solution:

1. **Identify the Radii of the Circles:**
Let the radius of circle \( C_1 \) be \( r_1 = 2 \) and the radius of circle \( C_2 \) be \( r_2 = 4 \).

2. **Use the Tangent Length Formula:**
The lengths of the tangents from point O to the circles can be expressed in terms of the radii. The lengths of the tangents \( OP \) and \( OQ \) from point O to circles \( C_1 \) and \( C_2 \) respectively can be given by:
\[ OP = \frac{r_1}{r_1 + r_2} \quad \text{and} \quad OQ = \frac{r_2}{r_1 + r_2} \]

3. **Calculate the Lengths of the Tangents:**
Substitute the values of \( r_1 \) and \( r_2 \):
\[ OP = \frac{2}{2 + 4} = \frac{2}{6} = \frac{1}{3} \]
\[ OQ = \frac{4}{2 + 4} = \frac{4}{6} = \frac{2}{3} \]

4. **Determine the Ratio of the Areas of the Triangles:**
The areas of triangles \( OAB \) and \( OCD \) formed by the tangents and the chord of contacts are proportional to the squares of the lengths of the tangents:
\[ \frac{\text{Area of } \triangle OAB}{\text{Area of } \triangle OCD} = \left(\frac{OQ}{OP}\right)^2 \]

5. **Calculate the Ratio:**
Substitute the lengths of the tangents:
\[ \frac{\text{Area of } \triangle OAB}{\text{Area of } \triangle OCD} = \left(\frac{\frac{2}{3}}{\frac{1}{3}}\right)^2 = \left(2\right)^2 = 4 \]

6. **Final Result:**
Thus, the ratio of the areas of triangles formed by the tangents drawn from O to the circles \( C_1 \) and \( C_2 \) and the chord of contacts of O with respect to the circles \( C_1 \) and \( C_2 \) respectively is:
\[ \text{Ratio} = 4 \]

### Conclusion:
The answer is option 3, which is 4 units.