Prove that the points `A(4,3),B(3,0)and C(2,-3)` are collinear. Also, find the line passing through these three points.

Answer

To prove that the points A(4, 3), B(3, 0), and C(2, -3) are collinear, we can follow these steps:

### Step 1: Find the slope of line AB
The slope \( m_{AB} \) between points A and B can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, for points A(4, 3) and B(3, 0):
- \( y_1 = 3 \), \( y_2 = 0 \)
- \( x_1 = 4 \), \( x_2 = 3 \)

Substituting these values into the formula:
\[ m_{AB} = \frac{0 - 3}{3 - 4} = \frac{-3}{-1} = 3 \]

### Step 2: Find the slope of line BC
Next, we find the slope \( m_{BC} \) between points B and C:
For points B(3, 0) and C(2, -3):
- \( y_1 = 0 \), \( y_2 = -3 \)
- \( x_1 = 3 \), \( x_2 = 2 \)

Substituting these values into the slope formula:
\[ m_{BC} = \frac{-3 - 0}{2 - 3} = \frac{-3}{-1} = 3 \]

### Step 3: Compare the slopes
Since \( m_{AB} = 3 \) and \( m_{BC} = 3 \), we have:
\[ m_{AB} = m_{BC} \]
This implies that the points A, B, and C are collinear.

### Step 4: Find the equation of the line passing through points A and B
We can use the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
Using point A(4, 3) and the slope \( m_{AB} = 3 \):
\[ y - 3 = 3(x - 4) \]
Expanding this:
\[ y - 3 = 3x - 12 \]
Rearranging gives:
\[ 3x - y - 9 = 0 \]

### Step 5: Verify point C lies on the line
To confirm that point C(2, -3) lies on the line, substitute \( x = 2 \) into the line equation:
\[ 3(2) - (-3) - 9 = 0 \]
Calculating:
\[ 6 + 3 - 9 = 0 \]
This simplifies to:
\[ 0 = 0 \]
Thus, point C lies on the line.

### Conclusion
Therefore, the points A(4, 3), B(3, 0), and C(2, -3) are collinear, and the equation of the line passing through these points is:
\[ 3x - y - 9 = 0 \]

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