Find the condition that the lines `x=py +q, z=ry +s and x=p'y +q', z= r'y+s'` may be perpendicular to each other.

Answer

To find the condition under which the lines given by the equations \(x = py + q, z = ry + s\) and \(x = p'y + q', z = r'y + s'\) are perpendicular to each other, we can follow these steps:

### Step 1: Rewrite the equations in parametric form
We start with the two sets of equations:

1. \(x = py + q\)
2. \(z = ry + s\)

From these, we can express \(y\) in terms of \(x\) and \(z\):
- From the first equation, we can express \(y\) as:
\[ y = \frac{x - q}{p} \]

- From the second equation, we can express \(y\) as:
\[ y = \frac{z - s}{r} \]

### Step 2: Set up the parametric equations
We can set up the parametric equations for the first line:
\[ \frac{x - q}{p} = y = \frac{z - s}{r} \]

For the second line:
1. \(x = p'y + q'\)
2. \(z = r'y + s'\)

Similarly, we can express \(y\) as:
- From the first equation:
\[ y = \frac{x - q'}{p'} \]

- From the second equation:
\[ y = \frac{z - s'}{r'} \]

### Step 3: Write the equations in symmetric form
The symmetric form of the lines can be expressed as:
For the first line:
\[ \frac{x - q}{p} = y = \frac{z - s}{r} \]

For the second line:
\[ \frac{x - q'}{p'} = y = \frac{z - s'}{r'} \]

### Step 4: Identify direction ratios
From the symmetric forms, we can identify the direction ratios of the lines:
- For the first line, the direction ratios are \(l_1 = p, m_1 = 1, n_1 = r\).
- For the second line, the direction ratios are \(l_2 = p', m_2 = 1, n_2 = r'\).

### Step 5: Condition for perpendicularity
Two lines are perpendicular if the following condition holds:
\[ l_1 \cdot l_2 + m_1 \cdot m_2 + n_1 \cdot n_2 = 0 \]

Substituting the values we found:
\[ p \cdot p' + 1 \cdot 1 + r \cdot r' = 0 \]

### Step 6: Simplify the equation
This simplifies to:
\[ p p' + 1 + r r' = 0 \]
or
\[ p p' + r r' = -1 \]

### Final Condition
Thus, the condition for the lines to be perpendicular is:
\[ p p' + r r' = -1 \]