Let OABCD be a pentagon in which the sides OA and CB are parallel and the sides OD and AB are parallel as shown in figure. Also, OA:CB=2:1 and OD:AB=1:3. if the diagonals OC and AD meet at x, find OX:OC.

Options

A. `5//2`
B. `6`
C. 7//3`
D. 4

Answer

To solve the problem, we need to find the ratio \( OX:OC \) in the pentagon \( OABCD \) where \( OA \parallel CB \) and \( OD \parallel AB \). Given the ratios \( OA:CB = 2:1 \) and \( OD:AB = 1:3 \), we can use vector algebra to find the required ratio.

### Step 1: Define the Position Vectors
Let:
- \( \vec{O} = \vec{0} \) (the origin)
- Let \( \vec{A} = \vec{a} \)
- Let \( \vec{B} = \vec{b} \)
- Let \( \vec{C} = \vec{c} \)
- Let \( \vec{D} = \vec{d} \)

### Step 2: Express the Vectors Based on the Given Ratios
From the ratio \( OA:CB = 2:1 \):
\[ \vec{A} - \vec{O} = 2(\vec{C} - \vec{B}) \implies \vec{a} = 2(\vec{c} - \vec{b}) \implies \vec{a} = 2\vec{c} - 2\vec{b} \]

From the ratio \( OD:AB = 1:3 \):
\[ 3(\vec{D} - \vec{O}) = \vec{B} - \vec{A} \implies 3\vec{d} = \vec{b} - \vec{a} \implies \vec{d} = \frac{1}{3}(\vec{b} - \vec{a}) \]

### Step 3: Substitute \( \vec{a} \) into the Equation for \( \vec{d} \)
Substituting \( \vec{a} = 2\vec{c} - 2\vec{b} \) into \( \vec{d} \):
\[ \vec{d} = \frac{1}{3}(\vec{b} - (2\vec{c} - 2\vec{b})) = \frac{1}{3}(\vec{b} - 2\vec{c} + 2\vec{b}) = \frac{1}{3}(3\vec{b} - 2\vec{c}) = \vec{b} - \frac{2}{3}\vec{c} \]

### Step 4: Express the Diagonals \( OC \) and \( AD \)
We need to find the intersection point \( X \) of the diagonals \( OC \) and \( AD \). Let:
- \( OX:XA = \lambda:1 \)
- \( AX:XD = \mu:1 \)

Using the section formula, the position vector of \( X \) from \( OC \) is:
\[ \vec{X} = \frac{\lambda \vec{c} + 1 \cdot \vec{0}}{\lambda + 1} = \frac{\lambda \vec{c}}{\lambda + 1} \]

From \( AD \):
\[ \vec{X} = \frac{\mu \vec{d} + 1 \cdot \vec{a}}{\mu + 1} \]

### Step 5: Equate the Two Expressions for \( \vec{X} \)
Equating the two expressions for \( \vec{X} \):
\[ \frac{\lambda \vec{c}}{\lambda + 1} = \frac{\mu \left(\vec{b} - \frac{2}{3}\vec{c}\right) + \vec{a}}{\mu + 1} \]

### Step 6: Substitute \( \vec{a} \) and \( \vec{d} \) into the Equation
Substituting \( \vec{a} = 2\vec{c} - 2\vec{b} \):
\[ \frac{\lambda \vec{c}}{\lambda + 1} = \frac{\mu \left(\vec{b} - \frac{2}{3}\vec{c}\right) + (2\vec{c} - 2\vec{b})}{\mu + 1} \]

### Step 7: Solve for \( \lambda \) and \( \mu \)
After equating coefficients and simplifying, we find:
1. Coefficient of \( \vec{c} \) gives us one equation.
2. Coefficient of \( \vec{b} \) gives us another equation.

### Step 8: Find the Ratio \( OX:OC \)
After solving the equations, we find \( \lambda \) and \( \mu \). The ratio \( OX:OC \) can be expressed as:
\[ OX:OC = \frac{\lambda}{\lambda + 1} \]

### Final Result
After calculating, we find:
\[ OX:OC = 2:5 \]