If points `A(60 hat i+3 hat j),\ B(40 hat i-8 hat j)` and `C(a hat i-52 hat j)` are collinear, then a is equal to

Options

A. a) `40`
B. b) `-40`
C. c) `20`
D. d) `-20`

Explanation

To find the value of \( a \) such that the points \( A(60 \hat{i} + 3 \hat{j}) \), \( B(40 \hat{i} - 8 \hat{j}) \), and \( C(a \hat{i} - 52 \hat{j}) \) are collinear, we can follow these steps:

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

### Step 2: Find the vector \( \vec{AB} \)
The vector \( \vec{AB} \) is given by:
\[ \vec{AB} = \vec{B} - \vec{A} = (40 \hat{i} - 8 \hat{j}) - (60 \hat{i} + 3 \hat{j}) = (40 - 60) \hat{i} + (-8 - 3) \hat{j} = -20 \hat{i} - 11 \hat{j} \]

### Step 3: Find the vector \( \vec{BC} \)
The vector \( \vec{BC} \) is given by:
\[ \vec{BC} = \vec{C} - \vec{B} = (a \hat{i} - 52 \hat{j}) - (40 \hat{i} - 8 \hat{j}) = (a - 40) \hat{i} + (-52 + 8) \hat{j} = (a - 40) \hat{i} - 44 \hat{j} \]

### Step 4: Set up the collinearity condition
Since points \( A \), \( B \), and \( C \) are collinear, the vectors \( \vec{AB} \) and \( \vec{BC} \) must be parallel. This means:
\[ \vec{AB} = \lambda \vec{BC} \]
for some scalar \( \lambda \).

### Step 5: Write the equation
Substituting the vectors we found:
\[ -20 \hat{i} - 11 \hat{j} = \lambda ((a - 40) \hat{i} - 44 \hat{j}) \]

### Step 6: Equate the coefficients
From the above equation, we can equate the coefficients of \( \hat{i} \) and \( \hat{j} \):
1. For \( \hat{i} \):
\[ -20 = \lambda (a - 40) \]
2. For \( \hat{j} \):
\[ -11 = \lambda (-44) \]

### Step 7: Solve for \( \lambda \)
From the second equation:
\[ \lambda = \frac{-11}{-44} = \frac{1}{4} \]

### Step 8: Substitute \( \lambda \) back to find \( a \)
Substituting \( \lambda \) into the first equation:
\[ -20 = \frac{1}{4} (a - 40) \]
Multiplying both sides by 4:
\[ -80 = a - 40 \]
Thus, solving for \( a \):
\[ a = -80 + 40 = -40 \]

### Final Answer
The value of \( a \) is \( -40 \).