The two lines x= ay+b, z=cy+d and `x=a^(1)y+b^(1), y=c^(1) y+d^(1)` are at right angles if a) `aa^(1)+c c^(1)=-1` b)`aa^(1)+b b^(1)+c c^(1)+1=0` c)`aa^(1)+b b^(1)+c c^(1)=0` d)`(a+a^(1))(b+b^(1))+c+c^(1)=0`

Options

A. `aa^(1)+ cc^(1)=-1`
B. `aa^(1)+b b^(1)+c c^(1)+1=0`
C. `aa^(1)+b b^(1)+c c^(1)=0`
D. `(a+a^(1))(b+b^(1))+c+c^(1)=0`

Answer

To determine the condition under which the two lines given by the equations \( x = ay + b, z = cy + d \) and \( x = a^{(1)}y + b^{(1)}, y = c^{(1)}y + d^{(1)} \) are at right angles, we can follow these steps:

### Step 1: Identify the Direction Ratios
The given lines can be expressed in a parametric form to identify their direction ratios.

For the first line:
- We can express it as:
\[ \frac{x - b}{a} = \frac{y - y_1}{1} = \frac{z - d}{c} \]
Here, the direction ratios are \( (a, 1, c) \).

For the second line:
- We can express it as:
\[ \frac{x - b^{(1)}}{a^{(1)}} = \frac{y - y_1}{1} = \frac{z - d^{(1)}}{c^{(1)}} \]
Here, the direction ratios are \( (a^{(1)}, 1, c^{(1)}) \).

### Step 2: Use the Condition for Perpendicular Lines
For two lines to be perpendicular, the dot product of their direction ratios must equal zero. Therefore, we have:
\[ a \cdot a^{(1)} + 1 \cdot 1 + c \cdot c^{(1)} = 0 \]
This simplifies to:
\[ aa^{(1)} + 1 + cc^{(1)} = 0 \]

### Step 3: Rearrange the Equation
Rearranging the equation gives us:
\[ aa^{(1)} + cc^{(1)} = -1 \]

### Conclusion
Thus, the condition under which the two lines are at right angles is:
\[ aa^{(1)} + cc^{(1)} = -1 \]
This corresponds to option (a).

### Final Answer
The correct answer is:
**a) \( aa^{(1)} + cc^{(1)} = -1 \)**

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