Equation of line passing through the points (1,2,3) ,(2,-1,2) is

Options

A. `(x-2)/(2) =(y-2)/(-1) =(z-3)/(2)`
B. `(x-1)/(1)=(y-2)/(-3)=(z-3)/(-1)`
C. `(x-2)/(1)=(y+1)/(2)=(z-1)/(3)`
D. none

Answer

To find the equation of the line passing through the points \( (1, 2, 3) \) and \( (2, -1, 2) \), we can use the symmetric form of the equation of a line in 3D space. Here are the steps to derive the equation:

### Step 1: Identify the Points
Let the points be:
- \( P_1 = (x_1, y_1, z_1) = (1, 2, 3) \)
- \( P_2 = (x_2, y_2, z_2) = (2, -1, 2) \)

### Step 2: Calculate the Direction Ratios
The direction ratios of the line can be found by subtracting the coordinates of the two points:
- Direction ratios \( (a, b, c) = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \)
- Calculate:
- \( a = 2 - 1 = 1 \)
- \( b = -1 - 2 = -3 \)
- \( c = 2 - 3 = -1 \)

So, the direction ratios are \( (1, -3, -1) \).

### Step 3: Write the Symmetric Form of the Line
The symmetric form of the line passing through the points is given by:
\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \]
Substituting the values:
\[ \frac{x - 1}{1} = \frac{y - 2}{-3} = \frac{z - 3}{-1} \]

### Step 4: Simplify the Equation
This can be simplified to:
\[ x - 1 = \frac{y - 2}{-3} = \frac{z - 3}{-1} \]
Thus, the final symmetric equation of the line is:
\[ x - 1 = \frac{y - 2}{-3} = \frac{z - 3}{-1} \]

### Final Answer
The equation of the line passing through the points \( (1, 2, 3) \) and \( (2, -1, 2) \) is:
\[ \frac{x - 1}{1} = \frac{y - 2}{-3} = \frac{z - 3}{-1} \]