If position vectors of the points A, B and C are a, b and c respectively and the points D and E divides line segment AC and AB in the ratio 2:1 and 1:3, respectively. Then, the points of intersection of BD and EC divides EC in the ratio

Options

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

Answer

To solve the problem, we need to determine the ratio in which the point of intersection \( M \) of the lines \( BD \) and \( EC \) divides the segment \( EC \). We will use the section formula and the information given about the points \( D \) and \( E \).

### Step-by-Step Solution:

1. **Identify the Position Vectors:**
- Let the position vectors of points \( A, B, C \) be \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) respectively.

2. **Find the Position Vector of Point \( D \):**
- Point \( D \) divides line segment \( AC \) in the ratio \( 2:1 \).
- Using the section formula, the position vector of \( D \) is given by:
\[ \mathbf{d} = \frac{2\mathbf{c} + 1\mathbf{a}}{2 + 1} = \frac{2\mathbf{c} + \mathbf{a}}{3} \]

3. **Find the Position Vector of Point \( E \):**
- Point \( E \) divides line segment \( AB \) in the ratio \( 1:3 \).
- Using the section formula, the position vector of \( E \) is given by:
\[ \mathbf{e} = \frac{1\mathbf{b} + 3\mathbf{a}}{1 + 3} = \frac{\mathbf{b} + 3\mathbf{a}}{4} \]

4. **Assume the Position Vector of Point \( M \):**
- Let \( M \) divide \( EC \) in the ratio \( x:1 \).
- Therefore, the position vector of \( M \) can be expressed as:
\[ \mathbf{m} = \frac{x\mathbf{c} + 1\mathbf{e}}{x + 1} \]

5. **Substituting the Position Vector of \( E \):**
- Substitute \( \mathbf{e} \) into the equation for \( \mathbf{m} \):
\[ \mathbf{m} = \frac{x\mathbf{c} + \frac{\mathbf{b} + 3\mathbf{a}}{4}}{x + 1} \]
- Simplifying gives:
\[ \mathbf{m} = \frac{4x\mathbf{c} + \mathbf{b} + 3\mathbf{a}}{4(x + 1)} \]

6. **Assume the Position Vector of Point \( M \) from Line \( BD \):**
- Let \( M \) divide \( BD \) in the ratio \( y:1 \).
- Thus, the position vector of \( M \) can also be expressed as:
\[ \mathbf{m} = \frac{y\mathbf{b} + 1\mathbf{d}}{y + 1} \]
- Substitute \( \mathbf{d} \):
\[ \mathbf{m} = \frac{y\mathbf{b} + \frac{2\mathbf{c} + \mathbf{a}}{3}}{y + 1} \]
- Simplifying gives:
\[ \mathbf{m} = \frac{3y\mathbf{b} + 2\mathbf{c} + \mathbf{a}}{3(y + 1)} \]

7. **Setting the Two Expressions for \( \mathbf{m} \) Equal:**
- Since both expressions represent the same point \( M \):
\[ \frac{4x\mathbf{c} + \mathbf{b} + 3\mathbf{a}}{4(x + 1)} = \frac{3y\mathbf{b} + 2\mathbf{c} + \mathbf{a}}{3(y + 1)} \]

8. **Cross-Multiplying to Eliminate Denominators:**
- Cross-multiplying gives:
\[ 3(y + 1)(4x\mathbf{c} + \mathbf{b} + 3\mathbf{a}) = 4(x + 1)(3y\mathbf{b} + 2\mathbf{c} + \mathbf{a}) \]

9. **Comparing Coefficients:**
- By comparing coefficients of \( \mathbf{c}, \mathbf{b}, \mathbf{a} \) on both sides, we can derive equations to solve for \( x \) and \( y \).

10. **Solving for the Ratio \( x:1 \):**
- After solving the equations, we find the value of \( x \).
- The final ratio in which \( M \) divides \( EC \) will be \( \frac{3}{2} : 1 \), which simplifies to \( 3:2 \).

### Final Answer:
The point of intersection \( M \) divides \( EC \) in the ratio \( 3:2 \).