Let O be an interior point of triangle ABC, such that `2vec(OA)+3vec(OB)+4vec(OC)=0`, then the ratio of the area of `DeltaABC` to the area of `DeltaAOC` is

Options

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

Answer

To solve the problem, we need to find the ratio of the area of triangle ABC to the area of triangle AOC given the condition \(2\vec{OA} + 3\vec{OB} + 4\vec{OC} = 0\).

### Step 1: Express the position vectors
Let the position vectors of points A, B, C, and O be represented as \(\vec{A}\), \(\vec{B}\), \(\vec{C}\), and \(\vec{O}\) respectively. The given equation can be rewritten as:
\[ 2\vec{O} = -3\vec{B} - 4\vec{C} \]
From this, we can express \(\vec{O}\) in terms of \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\).

### Step 2: Find the area of triangle AOC
The area of triangle AOC can be calculated using the formula:
\[ \text{Area}_{AOC} = \frac{1}{2} |\vec{OA} \times \vec{OC}| \]
Here, \(\vec{OA} = \vec{A} - \vec{O}\) and \(\vec{OC} = \vec{C} - \vec{O}\).

### Step 3: Find the area of triangle ABC
The area of triangle ABC can be calculated using the formula:
\[ \text{Area}_{ABC} = \frac{1}{2} |\vec{AB} \times \vec{AC}| \]
Where \(\vec{AB} = \vec{B} - \vec{A}\) and \(\vec{AC} = \vec{C} - \vec{A}\).

### Step 4: Use the given condition
From the given condition \(2\vec{OA} + 3\vec{OB} + 4\vec{OC} = 0\), we can find the coefficients that relate the areas of the triangles. The coefficients 2, 3, and 4 correspond to the areas of triangles AOC, BOC, and AOB respectively.

### Step 5: Calculate the ratio of areas
The ratio of the area of triangle ABC to the area of triangle AOC can be derived from the coefficients:
\[ \text{Area}_{ABC} = \text{Area}_{AOB} + \text{Area}_{AOC} + \text{Area}_{BOC} \]
Given that the areas are proportional to the coefficients, we can express:
\[ \text{Area}_{AOB} : \text{Area}_{AOC} : \text{Area}_{BOC} = 2 : 4 : 3 \]
Thus, the area of triangle AOC is \(4k\) and the area of triangle ABC is \(2k + 4k + 3k = 9k\).

### Step 6: Final Ratio
The ratio of the area of triangle ABC to the area of triangle AOC is:
\[ \frac{\text{Area}_{ABC}}{\text{Area}_{AOC}} = \frac{9k}{4k} = \frac{9}{4} \]

### Conclusion
Thus, the ratio of the area of triangle ABC to the area of triangle AOC is:
\[ \boxed{\frac{9}{4}} \]