Question

If the lines \(\rm {x -x_1\over a}={y-y_1\over b}={z-z_1\over c}\) and \(\rm {x -x_2\over p}={y-y_2\over q}={z-z_2\over r}\) are lies on a plane, then 

1. \(\begin{vmatrix} x_1& y_1& z_1\\ a& b &c \\ p& q &r \end{vmatrix} = 0\)
2. \(\begin{vmatrix} x_1-x_2& y_1-y_2 & z_1-z_2\\ a& b &c \\ p& q &r \end{vmatrix} = 0\)
3. \(\begin{vmatrix} x_1-x_2& y_1-y_2 & z_1-z_2\\ a& b &c \\ p& q &r \end{vmatrix} = 1\)
4. None of these

Answer 1

Correct Answer - Option 2 : \(\begin{vmatrix} x_1-x_2& y_1-y_2 & z_1-z_2\\ a& b &c \\ p& q &r \end{vmatrix} = 0\)

Concept:

Coplanar: Lines are said to be coplanar if they lie in the same plane.

If the two lines \(\rm {x -x_1\over a}={y-y_1\over b}={z-z_1\over c}\) and \(\rm {x -x_2\over p}={y-y_2\over q}={z-z_2\over r}\) are coplanar, then 

\(\begin{vmatrix} x_1-x_2& y_1-y_2 & z_1-z_2\\ a& b &c \\ p& q &r \end{vmatrix} = 0\)

∴ Option 2 is correct