If the points `A(lambda, 2lambda), B(3lambda,3lambda) and C(3,1)` are collinear, then `lambda=`

Options

A. `1//3`
B. `-1//3`
C. `2//3`
D. `2//3`

Explanation

To find the value of \( \lambda \) such that the points \( A(\lambda, 2\lambda) \), \( B(3\lambda, 3\lambda) \), and \( C(3, 1) \) are collinear, we can use the property that the area of the triangle formed by these three points must be zero.

### Step-by-Step Solution:

1. **Set Up the Area Formula**:
The area \( A \) of a triangle formed by points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Since the points are collinear, the area will be zero:
\[ 0 = \frac{1}{2} \left| \lambda(3\lambda - 1) + 3\lambda(1 - 2\lambda) + 3(2\lambda - 3\lambda) \right| \]

2. **Substituting the Points**:
Substitute \( A(\lambda, 2\lambda) \), \( B(3\lambda, 3\lambda) \), and \( C(3, 1) \) into the area formula:
\[ 0 = \frac{1}{2} \left| \lambda(3\lambda - 1) + 3\lambda(1 - 2\lambda) + 3(2\lambda - 3\lambda) \right| \]

3. **Simplifying the Expression**:
Simplifying the expression inside the absolute value:
\[ = \lambda(3\lambda - 1) + 3\lambda(1 - 2\lambda) + 3(2\lambda - 3\lambda) \]
\[ = \lambda(3\lambda - 1) + 3\lambda - 6\lambda^2 + 6\lambda - 9\lambda \]
\[ = 3\lambda^2 - \lambda + 3\lambda - 6\lambda^2 - 9\lambda \]
\[ = -3\lambda^2 - 7\lambda \]

4. **Setting the Determinant to Zero**:
Since the area is zero, we set the expression to zero:
\[ -3\lambda^2 - 7\lambda = 0 \]

5. **Factoring the Equation**:
Factor out \( \lambda \):
\[ \lambda(-3\lambda - 7) = 0 \]
This gives us two possible solutions:
\[ \lambda = 0 \quad \text{or} \quad -3\lambda - 7 = 0 \]

6. **Solving for \( \lambda \)**:
Solving \( -3\lambda - 7 = 0 \):
\[ -3\lambda = 7 \implies \lambda = -\frac{7}{3} \]

7. **Final Values**:
The values of \( \lambda \) are:
\[ \lambda = 0 \quad \text{or} \quad \lambda = -\frac{7}{3} \]

### Final Answer:
Thus, the values of \( \lambda \) such that the points \( A \), \( B \), and \( C \) are collinear are:
\[ \lambda = 0 \quad \text{or} \quad \lambda = -\frac{7}{3} \]