Question

If the lines $x=ay+b, z=cy+d$ and $x=a'y+b', z=c'y+d'$ are perpendicular, then:

• A. $aa'+cc'=1$
• B. $aa'+cc'=0$
• C. $aa'+cc'=-1$
• D. $aa'=cc'$

Hint:

For parametric lines, always extract direction ratios before applying perpendicular or angle conditions.

Answer 1

The Correct Option is B

Concept: A line in parametric form represents a direction vector. Two lines are perpendicular if the dot product of their direction vectors is zero.

Step 1: {Find direction vectors.}
Given: \[ x=ay+b,\quad z=cy+d \] So direction ratios are: \[ \vec{d_1} = (a,\,1,\,c) \] Similarly: \[ \vec{d_2} = (a',\,1,\,c') \]

Step 2: {Apply perpendicular condition.}
\[ \vec{d_1}\cdot \vec{d_2} = 0 \]

Step 3: {Compute dot product.}
\[ aa' + 1\cdot 1 + cc' = 0 \] \[ aa' + cc' + 1 = 0 \]

Step 4: {Rearrange.}
\[ aa' + cc' = -1 \]