The direction ratios of the line of intersection of the planes `x-y+z+3 =0 and x-3y -5 =0` are

Options

A. `lt 3,1,-2 gt`
B. `lt 3, -1,2 gt`
C. `lt 2, -4,1 gt`
D. `lt 2, 4,-1 gt`

Answer

To find the direction ratios of the line of intersection of the given planes \( x - y + z + 3 = 0 \) and \( x - 3y - 5 = 0 \), we can follow these steps:

### Step 1: Write the equations of the planes
The equations of the planes are:
1. Plane 1: \( x - y + z + 3 = 0 \)
2. Plane 2: \( x - 3y - 5 = 0 \)

### Step 2: Rearrange the equations
We can rearrange the equations to express \( z \) in terms of \( x \) and \( y \):
- From Plane 1:
\[ z = -x + y - 3 \]
- From Plane 2:
\[ x = 3y + 5 \]

### Step 3: Substitute \( x \) from Plane 2 into Plane 1
Now, we substitute \( x = 3y + 5 \) into the equation for \( z \):
\[ z = -(3y + 5) + y - 3 \]
Simplifying this gives:
\[ z = -3y - 5 + y - 3 = -2y - 8 \]

### Step 4: Express \( y \) in terms of \( z \)
From the equation \( z = -2y - 8 \), we can express \( y \) in terms of \( z \):
\[ 2y = -z - 8 \implies y = -\frac{z + 8}{2} \]

### Step 5: Express \( x \) in terms of \( y \)
Now we can substitute \( y \) back into the equation for \( x \):
\[ x = 3\left(-\frac{z + 8}{2}\right) + 5 \]
This simplifies to:
\[ x = -\frac{3(z + 8)}{2} + 5 = -\frac{3z + 24}{2} + 5 = -\frac{3z + 24}{2} + \frac{10}{2} = -\frac{3z + 14}{2} \]

### Step 6: Write the parametric equations
Now we can write the parametric equations in terms of \( z \):
1. \( x = -\frac{3z + 14}{2} \)
2. \( y = -\frac{z + 8}{2} \)
3. \( z = z \)

### Step 7: Identify the direction ratios
To find the direction ratios, we can express the equations in a standard form:
Let \( z = t \) (parameter). Then:
- \( x = -\frac{3t + 14}{2} \)
- \( y = -\frac{t + 8}{2} \)

The direction ratios can be obtained from the coefficients of \( t \):
- For \( x \): Coefficient is \( -\frac{3}{2} \)
- For \( y \): Coefficient is \( -\frac{1}{2} \)
- For \( z \): Coefficient is \( 1 \)

Thus, the direction ratios are \( -3 : -1 : 2 \), which can be simplified to \( 3 : 1 : -2 \).

### Final Answer
The direction ratios of the line of intersection of the planes are \( 3 : 1 : -2 \).

---