यदि रेखा \(\rm {x -x_1\over a}={y-y_1\over b}={z-z_1\over c}\) और \(\rm {x -x_2\over p}={y-y_2\over q}={z-z_2\over r}\) एक तल पर हैं, तो निम्न में कौन-सा सही है?

  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. इनमें से कोई नहीं 

Answer (Detailed Solution Below)

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

समतलीय: रेखाओं को समतलीय तब कहा जाता है यदि वे समान तल में होते हैं। 

यदि दो रेखाएं \(\rm {x -x_1\over a}={y-y_1\over b}={z-z_1\over c}\) और \(\rm {x -x_2\over p}={y-y_2\over q}={z-z_2\over r}\) एक-दूसरे के समतलीय हैं, तो 

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

∴ विकल्प 2 सही है। 

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