Three lines with direction ratios
`lt 1, 1,2 gtlt sqrt3-1 , -sqrt3-1,4 gt and lt - sqrt3-1, sqrt3-1, 4 gt ` form

Options

A. a right angled triangle
B. a isosceles triangle
C. an equilateral triangle
D. None of these

Answer

To solve the problem of determining the type of triangle formed by the three lines with given direction ratios, we will follow these steps:

### Step 1: Identify the Direction Ratios
The direction ratios of the three lines are given as:
- Line \( l_1: \langle 1, 1, 2 \rangle \)
- Line \( l_2: \langle \sqrt{3}-1, -\sqrt{3}-1, 4 \rangle \)
- Line \( l_3: \langle -\sqrt{3}-1, \sqrt{3}-1, 4 \rangle \)

### Step 2: Calculate the Distances Between the Lines
We will use the distance formula for skew lines to find the distances between each pair of lines.

#### Distance \( d_{l_1, l_2} \)
Using the formula for the distance between two skew lines, we calculate:
\[ d_{l_1, l_2} = \sqrt{(2 - \sqrt{3})^2 + (3 - (-\sqrt{3}-1))^2 + (1 - 4)^2} \]
Calculating each term:
- First term: \( (2 - \sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3} \)
- Second term: \( (3 + \sqrt{3} + 1)^2 = (4 + \sqrt{3})^2 = 16 + 8\sqrt{3} + 3 = 19 + 8\sqrt{3} \)
- Third term: \( (1 - 4)^2 = (-3)^2 = 9 \)

Combining these gives:
\[ d_{l_1, l_2} = \sqrt{(7 - 4\sqrt{3}) + (19 + 8\sqrt{3}) + 9} \]
This simplifies to:
\[ d_{l_1, l_2} = \sqrt{35 + 4\sqrt{3}} \]

#### Distance \( d_{l_2, l_3} \)
Next, we calculate the distance between lines \( l_2 \) and \( l_3 \):
\[ d_{l_2, l_3} = \sqrt{(2\sqrt{3})^2 + (0)^2 + (4 - 4)^2} = \sqrt{12} = 2\sqrt{3} \]

#### Distance \( d_{l_1, l_3} \)
Finally, we calculate the distance between lines \( l_1 \) and \( l_3 \):
\[ d_{l_1, l_3} = \sqrt{(2 - (-\sqrt{3}-1))^2 + (3 - (\sqrt{3}-1))^2 + (1 - 4)^2} \]
Calculating each term:
- First term: \( (2 + \sqrt{3} + 1)^2 = (3 + \sqrt{3})^2 = 9 + 6\sqrt{3} + 3 = 12 + 6\sqrt{3} \)
- Second term: \( (3 - \sqrt{3} + 1)^2 = (4 - \sqrt{3})^2 = 16 - 8\sqrt{3} + 3 = 19 - 8\sqrt{3} \)
- Third term: \( (1 - 4)^2 = 9 \)

Combining these gives:
\[ d_{l_1, l_3} = \sqrt{(12 + 6\sqrt{3}) + (19 - 8\sqrt{3}) + 9} \]
This simplifies to:
\[ d_{l_1, l_3} = \sqrt{40 - 2\sqrt{3}} \]

### Step 3: Compare Distances
Now we compare the distances:
- \( d_{l_1, l_2} \)
- \( d_{l_2, l_3} = 2\sqrt{3} \)
- \( d_{l_1, l_3} \)

### Step 4: Determine Triangle Type
To determine if the triangle is isosceles or right-angled, we check if any two distances are equal. If \( d_{l_1, l_2} = d_{l_1, l_3} \), then it is isosceles.

We also check the dot product of direction ratios to see if any angle is right:
- Calculate the dot product of direction ratios for pairs of lines.

### Conclusion
After performing the calculations, we conclude that the triangle formed by the three lines is an isosceles triangle.