Q54 / 90 · JEE Main 2019 (09 Jan Shift 2) · Mathematics

If the lines $x = a y + b, z = c y + d$ and $x = a^{'} z + b^{'}, y = c^{'} z + d^{'}$ are perpendicular, then

Solution

Direction Ratios of the first line

|\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & - a & 0 \\ 0 & c & - 1 \end{matrix}| = a \hat{i} + \hat{j} + c \hat{k}

Hence, Direction Ratios of the first line are

a, 1, c .

For Direction Ratios of the second line

|\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & - a^{'} \\ 0 & 1 & - c^{'} \end{matrix}| = a^{'} \hat{i} + c^{'} \hat{j} + \hat{k}

Hence, Direction Ratios of the

2^{n d}

line

= a^{'}, c^{'}, 1

Since, both the lines are perpendicular,

a . a^{'} + c^{'} + c = 0

Hence, the correct answer is option: 'B'.