If the lines `x=a_(1)y + b_(1), z=c_(1)y +d_(1)` and `x=a_(2)y +b_(2), z=c_(2)y + d_(2)` are perpendicular, then

Options

A. `ac _(1) + a_(1) c =1`
B. ` a a _(1) + c c_(1) + 1=0`
C. ` bc _(1) + b_(1) c + 1=0`
D. None of these

Answer

To determine the condition for the lines given by the equations \(x = a_1y + b_1, z = c_1y + d_1\) and \(x = a_2y + b_2, z = c_2y + d_2\) to be perpendicular, we can follow these steps:

### Step 1: Write the equations in standard form
The equations of the lines can be rewritten in a standard form. For the first line:
\[ x - b_1 = a_1y \quad \text{and} \quad z - d_1 = c_1y \]
This can be expressed as:
\[ \frac{x - b_1}{a_1} = y = \frac{z - d_1}{c_1} \]

For the second line:
\[ x - b_2 = a_2y \quad \text{and} \quad z - d_2 = c_2y \]
This can be expressed as:
\[ \frac{x - b_2}{a_2} = y = \frac{z - d_2}{c_2} \]

### Step 2: Identify direction ratios
From the standard form, we can identify the direction ratios of the two lines:
- For the first line, the direction ratios are \(d_1 = (a_1, 1, c_1)\).
- For the second line, the direction ratios are \(d_2 = (a_2, 1, c_2)\).

### Step 3: Use the dot product condition
For two lines to be perpendicular, the dot product of their direction ratios must be zero:
\[ d_1 \cdot d_2 = 0 \]
Calculating the dot product:
\[ (a_1, 1, c_1) \cdot (a_2, 1, c_2) = a_1a_2 + 1 \cdot 1 + c_1c_2 = 0 \]
This simplifies to:
\[ a_1a_2 + 1 + c_1c_2 = 0 \]

### Step 4: Conclusion
Thus, the condition for the lines to be perpendicular is:
\[ a_1a_2 + c_1c_2 + 1 = 0 \]