If the point R(1, -2) divides externally the line segment joining P(2, 5) and Q in the ratio 3 : 4, what will be the coordinates of Q?

Options

A. (- 3, 6)
B. (2, - 4)
C. (3, 6)
D. (1, 2)

Answer

To find the coordinates of point Q, we will use the formula for external division of a line segment. Given that point R(1, -2) divides the line segment joining points P(2, 5) and Q externally in the ratio 3:4, we can denote the coordinates of Q as (x2, y2).

### Step-by-Step Solution:

1. **Identify the known points and the ratio**:
- Point P = (x1, y1) = (2, 5)
- Point R = (x, y) = (1, -2)
- Ratio of division = m:n = 3:4

2. **Use the formula for external division**:
The formula for the coordinates of a point that divides a line segment externally in the ratio m:n is given by:
\[ x = \frac{m \cdot x_2 - n \cdot x_1}{m - n} \]
\[ y = \frac{m \cdot y_2 - n \cdot y_1}{m - n} \]

3. **Set up the equations**:
For the x-coordinate:
\[ 1 = \frac{3 \cdot x_2 - 4 \cdot 2}{3 - 4} \]
Simplifying this gives:
\[ 1 = \frac{3x_2 - 8}{-1} \]
Therefore:
\[ -1 = 3x_2 - 8 \]
Rearranging gives:
\[ 3x_2 = 7 \implies x_2 = \frac{7}{3} \]

For the y-coordinate:
\[ -2 = \frac{3 \cdot y_2 - 4 \cdot 5}{3 - 4} \]
Simplifying this gives:
\[ -2 = \frac{3y_2 - 20}{-1} \]
Therefore:
\[ 2 = 3y_2 - 20 \]
Rearranging gives:
\[ 3y_2 = 22 \implies y_2 = \frac{22}{3} \]

4. **Combine the results**:
The coordinates of point Q are:
\[ Q = \left(\frac{7}{3}, \frac{22}{3}\right) \]

### Final Answer:
The coordinates of point Q are \(\left(\frac{7}{3}, \frac{22}{3}\right)\).