Under which one of the following conditions are the lines x= ay + b ;   z = cy + d and x = ey +f ;   z = gy + h perpendicular

Options

A. A) ae + cg -1=0
B. B) ae + bf - 1= 0
C. C) ae + cg + 1 =0
D. D) ag + ce + 1 = 0

Answer

To determine the conditions under which the lines given by the equations \(x = ay + b\), \(z = cy + d\) and \(x = ey + f\), \(z = gy + h\) are perpendicular, we need to find their direction ratios and then apply the condition for perpendicularity.

### Step-by-Step Solution:

1. **Identify the equations of the lines**:
- The first line is given by:
\[ x = ay + b \quad \text{and} \quad z = cy + d \]
- The second line is given by:
\[ x = ey + f \quad \text{and} \quad z = gy + h \]

2. **Rewrite the equations in symmetric form**:
- For the first line:
\[ \frac{x - b}{a} = \frac{y - 0}{1} = \frac{z - d}{c} \]
Thus, the direction ratios for the first line are \( (a, 1, c) \).

- For the second line:
\[ \frac{x - f}{e} = \frac{y - 0}{1} = \frac{z - h}{g} \]
Thus, the direction ratios for the second line are \( (e, 1, g) \).

3. **Apply the condition for perpendicularity**:
- Two lines are perpendicular if the product of their direction ratios equals zero. Therefore, we need to check:
\[ a \cdot e + 1 \cdot 1 + c \cdot g = 0 \]
- This simplifies to:
\[ ae + 1 + cg = 0 \]

4. **Rearranging the equation**:
- We can rearrange this to find the condition:
\[ ae + cg + 1 = 0 \]

5. **Conclusion**:
- The lines \(x = ay + b\), \(z = cy + d\) and \(x = ey + f\), \(z = gy + h\) are perpendicular if the condition \(ae + cg + 1 = 0\) holds true.