Find the co-ordinates of a point which divides the line segment joining the points (3, 0) and (0, 2) in the ratio 2 : 1 externally.
A. `(3,4)`
B. `(-3,4)`
C. `(3,-4)`
D.`(-3,-4)`

Answer

To find the coordinates of a point that divides the line segment joining the points (3, 0) and (0, 2) in the ratio 2:1 externally, we will use the formula for external division.

### Step 1: Identify the points and the ratio
Let the points be:
- Point A (x1, y1) = (3, 0)
- Point B (x2, y2) = (0, 2)

The ratio in which the point divides the line segment externally is given as 2:1.

### Step 2: Use the external division formula
The formula for the coordinates (x, y) of a point that divides the line segment joining two points (x1, y1) and (x2, y2) in the ratio m:n externally is given by:

\[ \left( \frac{m \cdot x_2 - n \cdot x_1}{m - n}, \frac{m \cdot y_2 - n \cdot y_1}{m - n} \right) \]

Here, m = 2 and n = 1.

### Step 3: Substitute the values into the formula
Substituting the values into the formula:

- For the x-coordinate:
\[ x = \frac{2 \cdot 0 - 1 \cdot 3}{2 - 1} = \frac{0 - 3}{1} = \frac{-3}{1} = -3 \]

- For the y-coordinate:
\[ y = \frac{2 \cdot 2 - 1 \cdot 0}{2 - 1} = \frac{4 - 0}{1} = \frac{4}{1} = 4 \]

### Step 4: Write the final coordinates
Thus, the coordinates of the point that divides the line segment externally in the ratio 2:1 are:
\[ (-3, 4) \]

### Step 5: Conclusion
Looking at the options provided, the correct answer is:
**B. (-3, 4)**