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प्रश्न
If the lines \[x =\] 5 , \[\frac{y}{3 - \alpha} = \frac{z}{- 2}\] and \[x = \alpha\] \[\frac{y}{- 1} = \frac{z}{2 - \alpha}\] are coplanar, find the values of \[\alpha\].
उत्तर
The lines \[\frac{x - x_1}{a_1} = \frac{y - y_1}{b_1} = \frac{z - z_1}{c_1}\] and \[\frac{x - x_2}{a_2} = \frac{y - y_2}{b_2} = \frac{z - z_2}{c_2}\] are coplanar if \[\begin{vmatrix}x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2\end{vmatrix} = 0\]
The given lines
\[\therefore \begin{vmatrix}\alpha - 5 & 0 - 0 & 0 - 0 \\ 0 & 3 - \alpha & - 2 \\ 0 & - 1 & 2 - \alpha\end{vmatrix} = 0\]
\[ \Rightarrow \begin{vmatrix}\alpha - 5 & 0 & 0 \\ 0 & 3 - \alpha & - 2 \\ 0 & - 1 & 2 - \alpha\end{vmatrix} = 0\]
\[ \Rightarrow \left( \alpha - 5 \right)\left[ \left( 3 - \alpha \right) \times \left( 2 - \alpha \right) - 2 \right] - 0 + 0 = 0\]
\[ \Rightarrow \left( \alpha - 5 \right)\left( \alpha^2 - 5\alpha + 4 \right) = 0\]
\[ \Rightarrow \left( \alpha - 5 \right)\left( \alpha - 1 \right)\left( \alpha - 4 \right) = 0\]
\[\Rightarrow \alpha - 1 = 0 \text{ or } \alpha - 4 = 0 \text{ or } \alpha - 5 = 0\]
\[ \Rightarrow \alpha = 1 \text { or } \alpha = 4 \text{ or } \alpha = 5\] Thus, the values of \[\alpha\] are 1, 4 and 5.
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