Equations of the line joining `(2,-3,1)and (3,-4,-5)` are

Options

A. `(x-2)/3 =(y+3)/(-4) = (z-1)/(-5)`
B. `(x-3)/2=(y+4)/(-3) (z+5)/1`
C. `2-x = y +3 = (z-1)/6`
D. `(x-2)/3 = (y+4)/(-3) = (z-1)/(-1)`

Answer

To find the equations of the line joining the points \( A(2, -3, 1) \) and \( B(3, -4, -5) \), we can use the parametric form of the line in space.

### Step-by-Step Solution:

1. **Identify the Points**:
We have two points:
- Point A: \( (x_1, y_1, z_1) = (2, -3, 1) \)
- Point B: \( (x_2, y_2, z_2) = (3, -4, -5) \)

2. **Calculate Direction Ratios**:
The direction ratios of the line can be found by subtracting the coordinates of point A from point B:
\[ \text{Direction ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) = (3 - 2, -4 - (-3), -5 - 1) = (1, -1, -6) \]

3. **Write the Parametric Equations**:
The parametric equations of the line can be expressed as:
\[ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1} \]
Substituting the values:
\[ \frac{x - 2}{1} = \frac{y + 3}{-1} = \frac{z - 1}{-6} \]

4. **Rearranging the Equations**:
We can rearrange the above equation to express it in a more standard form:
\[ x - 2 = \frac{y + 3}{-1} \quad \text{and} \quad x - 2 = \frac{z - 1}{-6} \]
This gives us:
\[ x - 2 = - (y + 3) \quad \text{and} \quad x - 2 = -\frac{1}{6}(z - 1) \]

5. **Final Form**:
To express it in a single equation, we can write:
\[ 2 - x = y + 3 = \frac{z - 1}{-6} \]
This can also be expressed as:
\[ 2 - x = -(y + 3) = -\frac{(z - 1)}{6} \]

### Summary of the Equations:
The equations of the line joining the points \( (2, -3, 1) \) and \( (3, -4, -5) \) can be summarized as:
\[ 2 - x = y + 3 = \frac{z - 1}{-6} \]