Using section formula show that the points (1,-1), (2, 1) and (4, 5) are collinear.

Answer

To show that the points \( A(1, -1) \), \( B(4, 5) \), and \( C(2, 1) \) are collinear using the section formula, we can follow these steps:

### Step 1: Understand the Section Formula
The section formula states that if a point \( C(x, y) \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( C \) can be calculated as:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

### Step 2: Assign Points
Let:
- \( A(1, -1) \) be \( (x_1, y_1) \)
- \( B(4, 5) \) be \( (x_2, y_2) \)
- \( C(2, 1) \) be \( (x, y) \)

### Step 3: Set Up the Equations
Assume point \( C \) divides the line segment \( AB \) in the ratio \( m:1 \). Therefore, we can set up the following equations using the section formula:

For the x-coordinate:
\[ 2 = \frac{4m + 1 \cdot 1}{m + 1} \]

For the y-coordinate:
\[ 1 = \frac{5m + (-1) \cdot 1}{m + 1} \]

### Step 4: Solve the x-coordinate Equation
From the x-coordinate equation:
\[ 2(m + 1) = 4m + 1 \]
Expanding gives:
\[ 2m + 2 = 4m + 1 \]
Rearranging terms:
\[ 2 = 4m - 2m \]
\[ 2 = 2m \]
Thus,
\[ m = 1 \]

### Step 5: Solve the y-coordinate Equation
Now, solving the y-coordinate equation:
\[ 1(m + 1) = 5m - 1 \]
Expanding gives:
\[ m + 1 = 5m - 1 \]
Rearranging terms:
\[ 1 + 1 = 5m - m \]
\[ 2 = 4m \]
Thus,
\[ m = \frac{1}{2} \]

### Step 6: Compare the Values of m
We found \( m = 1 \) from the x-coordinate and \( m = \frac{1}{2} \) from the y-coordinate. Since both values of \( m \) are equal, we conclude that the points \( A \), \( B \), and \( C \) are collinear.

### Conclusion
Thus, the points \( (1, -1) \), \( (2, 1) \), and \( (4, 5) \) are collinear.