Which of the following points will be collinear with the points (-3, 4) and (2, -5) ?

Options

A. (0, 0)
B. (7, -14)
C. (0, -1)
D. (3, 1)

Answer

To determine which of the given points is collinear with the points (-3, 4) and (2, -5), we will first find the equation of the line that passes through these two points.

### Step-by-Step Solution:

1. **Identify the Points**:
The two points we have are:
- Point A: (-3, 4)
- Point B: (2, -5)

2. **Find the Slope (m)**:
The slope \( m \) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values:
\[ m = \frac{-5 - 4}{2 - (-3)} = \frac{-9}{5} = -\frac{9}{5} \]

3. **Use Point-Slope Form to Find the Equation of the Line**:
The point-slope form of the equation of a line is:
\[ y - y_1 = m(x - x_1) \]
Using point A (-3, 4):
\[ y - 4 = -\frac{9}{5}(x + 3) \]
Simplifying this, we get:
\[ y - 4 = -\frac{9}{5}x - \frac{27}{5} \]
\[ y = -\frac{9}{5}x + 4 - \frac{27}{5} \]
\[ y = -\frac{9}{5}x + \frac{20}{5} - \frac{27}{5} \]
\[ y = -\frac{9}{5}x - \frac{7}{5} \]

4. **Check Each Point for Collinearity**:
We will check each of the given options to see if they satisfy the line equation \( y = -\frac{9}{5}x - \frac{7}{5} \).

- **Option A: (0, 0)**:
\[ y = -\frac{9}{5}(0) - \frac{7}{5} = -\frac{7}{5} \quad (\text{Not collinear}) \]

- **Option B: (7, -14)**:
\[ y = -\frac{9}{5}(7) - \frac{7}{5} = -\frac{63}{5} - \frac{7}{5} = -\frac{70}{5} = -14 \quad (\text{Collinear}) \]

- **Option C: (0, -1)**:
\[ y = -\frac{9}{5}(0) - \frac{7}{5} = -\frac{7}{5} \quad (\text{Not collinear}) \]

- **Option D: (3, 1)**:
\[ y = -\frac{9}{5}(3) - \frac{7}{5} = -\frac{27}{5} - \frac{7}{5} = -\frac{34}{5} \quad (\text{Not collinear}) \]

5. **Conclusion**:
The only point that is collinear with the points (-3, 4) and (2, -5) is **Option B: (7, -14)**.