The points with position vectors `10hati+3hatj,12hati-5hatj and ahati+11hatj` are collinear, if a is equal to

Options

A. `-8`
B. 4
C. 8
D. 12

Explanation

To determine the value of \( a \) such that the points with position vectors \( 10\hat{i} + 3\hat{j} \), \( 12\hat{i} - 5\hat{j} \), and \( a\hat{i} + 11\hat{j} \) are collinear, we can follow these steps:

### Step 1: Define the position vectors
Let:
- \( \vec{A} = 10\hat{i} + 3\hat{j} \)
- \( \vec{B} = 12\hat{i} - 5\hat{j} \)
- \( \vec{C} = a\hat{i} + 11\hat{j} \)

### Step 2: Find the vectors \( \vec{AB} \) and \( \vec{BC} \)
To find the vectors \( \vec{AB} \) and \( \vec{BC} \):
- \( \vec{AB} = \vec{B} - \vec{A} = (12\hat{i} - 5\hat{j}) - (10\hat{i} + 3\hat{j}) \)
- Simplifying this gives:
\[ \vec{AB} = (12 - 10)\hat{i} + (-5 - 3)\hat{j} = 2\hat{i} - 8\hat{j} \]

- Now, find \( \vec{BC} \):
\[ \vec{BC} = \vec{C} - \vec{B} = (a\hat{i} + 11\hat{j}) - (12\hat{i} - 5\hat{j}) \]
- Simplifying this gives:
\[ \vec{BC} = (a - 12)\hat{i} + (11 + 5)\hat{j} = (a - 12)\hat{i} + 16\hat{j} \]

### Step 3: Set up the collinearity condition
For the points to be collinear, the vectors \( \vec{AB} \) and \( \vec{BC} \) must be scalar multiples of each other. This means:
\[ \vec{AB} = p \cdot \vec{BC} \]
for some non-zero scalar \( p \).

### Step 4: Write the equations for the components
From the vectors:
\[ 2\hat{i} - 8\hat{j} = p \cdot ((a - 12)\hat{i} + 16\hat{j}) \]
This gives us two equations:
1. \( 2 = p(a - 12) \)
2. \( -8 = 16p \)

### Step 5: Solve for \( p \)
From the second equation:
\[ p = \frac{-8}{16} = -\frac{1}{2} \]

### Step 6: Substitute \( p \) back into the first equation
Substituting \( p \) into the first equation:
\[ 2 = -\frac{1}{2}(a - 12) \]
Multiplying both sides by -2 gives:
\[ -4 = a - 12 \]
Thus:
\[ a = 12 - 4 = 8 \]

### Conclusion
The value of \( a \) is \( 8 \).