If O (origin) is a point inside the triangle PQR such that `vec(OP)+k_(1)vec(OQ)+k_(2)vec(OR)=0`, where `k_(1), k_(2)` are constants such that `("Area"(DeltaPQR))/("Area"(DeltaOQR))=4`, then the value of `k_(1)+k_(2)` is :

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem, we need to find the value of \( k_1 + k_2 \) given the equation:

\[ \vec{OP} + k_1 \vec{OQ} + k_2 \vec{OR} = 0 \]

and the condition that the area of triangle \( PQR \) is four times the area of triangle \( OQR \).

### Step 1: Understand the Area Relationship
We know that:

\[ \frac{\text{Area}(\Delta PQR)}{\text{Area}(\Delta OQR)} = 4 \]

This implies that the area of triangle \( PQR \) is four times that of triangle \( OQR \).

### Step 2: Area of Triangles in Vector Form
The area of a triangle formed by vectors \( \vec{A} \), \( \vec{B} \), and \( \vec{C} \) can be expressed as:

\[ \text{Area} = \frac{1}{2} |\vec{AB} \times \vec{AC}| \]

For triangle \( PQR \), the area can be expressed as:

\[ \text{Area}(\Delta PQR) = \frac{1}{2} |\vec{PQ} \times \vec{PR}| \]

And for triangle \( OQR \):

\[ \text{Area}(\Delta OQR) = \frac{1}{2} |\vec{OQ} \times \vec{OR}| \]

### Step 3: Expressing the Area Ratio
Using the above expressions, we can write:

\[ \frac{\frac{1}{2} |\vec{PQ} \times \vec{PR}|}{\frac{1}{2} |\vec{OQ} \times \vec{OR}|} = 4 \]

This simplifies to:

\[ \frac{|\vec{PQ} \times \vec{PR}|}{|\vec{OQ} \times \vec{OR}|} = 4 \]

### Step 4: Substitute Vectors
Let:

- \( \vec{OP} = \vec{P} \)
- \( \vec{OQ} = \vec{Q} \)
- \( \vec{OR} = \vec{R} \)

Then:

\[ \vec{PQ} = \vec{Q} - \vec{P} \]
\[ \vec{PR} = \vec{R} - \vec{P} \]

### Step 5: Area Calculation
Now we can express the areas as:

\[ |\vec{PQ} \times \vec{PR}| = |(\vec{Q} - \vec{P}) \times (\vec{R} - \vec{P})| \]
\[ |\vec{OQ} \times \vec{OR}| = |\vec{Q} \times \vec{R}| \]

### Step 6: Area Ratio
Substituting these into the area ratio gives:

\[ \frac{|(\vec{Q} - \vec{P}) \times (\vec{R} - \vec{P})|}{|\vec{Q} \times \vec{R}|} = 4 \]

### Step 7: Expressing \( \vec{P} \)
From the original equation \( \vec{OP} + k_1 \vec{OQ} + k_2 \vec{OR} = 0 \), we can express \( \vec{P} \):

\[ \vec{P} = -k_1 \vec{Q} - k_2 \vec{R} \]

### Step 8: Substitute \( \vec{P} \) in Area Expression
Substituting \( \vec{P} \) into the area expression, we can simplify it further:

\[ |\vec{PQ} \times \vec{PR}| = |(\vec{Q} - (-k_1 \vec{Q} - k_2 \vec{R})) \times (\vec{R} - (-k_1 \vec{Q} - k_2 \vec{R}))| \]

### Step 9: Solve for \( k_1 + k_2 \)
After simplifying the expressions, we find:

\[ |(-k_1 \vec{Q} - k_2 \vec{R}) \times \vec{Q}| + |(-k_1 \vec{Q} - k_2 \vec{R}) \times \vec{R}| = 4 |\vec{Q} \times \vec{R}| \]

This leads us to the conclusion that:

\[ |k_1 + k_2 + 1| = 4 \]

Thus, we have two cases:

1. \( k_1 + k_2 + 1 = 4 \) leading to \( k_1 + k_2 = 3 \)
2. \( k_1 + k_2 + 1 = -4 \) leading to \( k_1 + k_2 = -5 \)

Since \( O \) is inside triangle \( PQR \), we take the first case.

### Final Answer
The value of \( k_1 + k_2 \) is:

\[ \boxed{3} \]