Equation of the line passing through the point (1, -2, 5) and having d.r's 1, 2, 3 is

Options

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

Answer

To find the equation of the line passing through the point (1, -2, 5) with direction ratios 1, 2, and 3, we can follow these steps:

### Step 1: Identify the point and direction ratios
We are given:
- Point \( P(x_1, y_1, z_1) = (1, -2, 5) \)
- Direction ratios \( d_x = 1, d_y = 2, d_z = 3 \)

### Step 2: Write the general equation of the line
The general form of the equation of a line in 3D is given by:
\[ \frac{x - x_1}{d_x} = \frac{y - y_1}{d_y} = \frac{z - z_1}{d_z} \]
Substituting the values we have:
\[ \frac{x - 1}{1} = \frac{y + 2}{2} = \frac{z - 5}{3} \]

### Step 3: Simplify the equation
Now we can express the equation in a more simplified manner:
\[ x - 1 = \frac{y + 2}{2} = \frac{z - 5}{3} \]
This can be rewritten as:
1. \( x - 1 = t \) (where \( t \) is a parameter)
2. \( y + 2 = 2t \)
3. \( z - 5 = 3t \)

From these equations, we can express \( y \) and \( z \) in terms of \( t \):
- \( y = 2t - 2 \)
- \( z = 3t + 5 \)

### Step 4: Write the parametric equations
Thus, the parametric equations of the line can be written as:
\[ x = 1 + t \]
\[ y = 2t - 2 \]
\[ z = 3t + 5 \]

### Step 5: Final equation of the line
The symmetric form of the equation of the line can be expressed as:
\[ \frac{x - 1}{1} = \frac{y + 2}{2} = \frac{z - 5}{3} \]

### Conclusion
The equation of the line passing through the point (1, -2, 5) with direction ratios 1, 2, and 3 is:
\[ \frac{x - 1}{1} = \frac{y + 2}{2} = \frac{z - 5}{3} \]