The points with position vectors `60hati+3hatj,40hati-8hatj, ahati-52hatj` are collinear if (A) `a=-40` (B) `a=40` (C) `a=20` (D) none of these

Options

A. `-40`
B. `40`
C. 20
D. none of these

Explanation

To determine the value of \( a \) for which the points with position vectors \( 60\hat{i} + 3\hat{j} \), \( 40\hat{i} - 8\hat{j} \), and \( a\hat{i} - 52\hat{j} \) are collinear, we can follow these steps:

### Step 1: Define the position vectors
Let:
- \( \vec{A} = 60\hat{i} + 3\hat{j} \)
- \( \vec{B} = 40\hat{i} - 8\hat{j} \)
- \( \vec{C} = a\hat{i} - 52\hat{j} \)

### Step 2: Use the collinearity condition
For points \( A \), \( B \), and \( C \) to be collinear, the vector \( \vec{C} - \vec{A} \) must be a scalar multiple of the vector \( \vec{B} - \vec{A} \). This can be expressed as:
\[ \vec{C} - \vec{A} = k(\vec{B} - \vec{A}) \]
for some scalar \( k \).

### Step 3: Calculate the vectors
Calculate \( \vec{C} - \vec{A} \):
\[ \vec{C} - \vec{A} = (a\hat{i} - 52\hat{j}) - (60\hat{i} + 3\hat{j}) = (a - 60)\hat{i} - (52 + 3)\hat{j} = (a - 60)\hat{i} - 55\hat{j} \]

Calculate \( \vec{B} - \vec{A} \):
\[ \vec{B} - \vec{A} = (40\hat{i} - 8\hat{j}) - (60\hat{i} + 3\hat{j}) = (40 - 60)\hat{i} - (8 + 3)\hat{j} = -20\hat{i} - 11\hat{j} \]

### Step 4: Set up the equation
Now we have:
\[ (a - 60)\hat{i} - 55\hat{j} = k(-20\hat{i} - 11\hat{j}) \]

### Step 5: Equate the components
From the above equation, equate the coefficients of \( \hat{i} \) and \( \hat{j} \):
1. For \( \hat{i} \):
\[ a - 60 = -20k \quad \text{(1)} \]
2. For \( \hat{j} \):
\[ -55 = -11k \quad \text{(2)} \]

### Step 6: Solve for \( k \)
From equation (2):
\[ -55 = -11k \implies k = \frac{55}{11} = 5 \]

### Step 7: Substitute \( k \) back into equation (1)
Now substitute \( k = 5 \) into equation (1):
\[ a - 60 = -20(5) \implies a - 60 = -100 \]
\[ a = -100 + 60 = -40 \]

### Conclusion
Thus, the value of \( a \) for which the points are collinear is:
\[ \boxed{-40} \]