What is the ratio in which point P(1, 2) divides the join of A(-2, 1) and B(7,4)?

Options

A. `1:2`
B. `2:1`
C. `3:4`
D. `2:3`

Answer

To find the ratio in which point P(1, 2) divides the line segment joining points A(-2, 1) and B(7, 4), we can use the section formula. The section formula states that if a point divides a line segment in the ratio m:n, then the coordinates of the point can be calculated using the following formulas:

\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

### Step-by-Step Solution:

1. **Identify the Coordinates:**
- Let A(-2, 1) be (x1, y1).
- Let B(7, 4) be (x2, y2).
- Let P(1, 2) be the point that divides the line segment.

2. **Assume the Ratio:**
- Let the ratio in which point P divides AB be k:1.

3. **Apply the Section Formula:**
- For the x-coordinate:
\[ 1 = \frac{7k + (-2) \cdot 1}{k + 1} \]
Simplifying this gives:
\[ 1 = \frac{7k - 2}{k + 1} \]
- For the y-coordinate:
\[ 2 = \frac{4k + 1 \cdot 1}{k + 1} \]
Simplifying this gives:
\[ 2 = \frac{4k + 1}{k + 1} \]

4. **Solve the x-coordinate Equation:**
- Cross-multiplying:
\[ 1(k + 1) = 7k - 2 \]
\[ k + 1 = 7k - 2 \]
Rearranging gives:
\[ 6k = 3 \implies k = \frac{3}{6} = \frac{1}{2} \]

5. **Determine the Ratio:**
- The ratio k:1 is thus \( \frac{1}{2}:1 \), which simplifies to 1:2.

### Final Answer:
The ratio in which point P(1, 2) divides the line segment joining A(-2, 1) and B(7, 4) is **1:2**.