If the lines \frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}\; and\; \frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}  are coplanar, then k can have :

  • Option 1)

    exactly three values

  • Option 2)

    any value

  • Option 3)

    exactly one values

  • Option 4)

    exactly two values

 

Answers (2)

As we learnt in

Condition for lines to be intersecting (cartesian form) -

Their shortest distance should be 0

Also the condition for coplanar lines

-

 

 Fo coplaner lines we have 

\begin{vmatrix} 1 &-1 &-1 \\ 1&1 &-k \\ k&2 &1 \end{vmatrix}= 0

1\left ( 1+2k \right )+1\left ( 1+k^{2} \right )-1\left ( 2-k \right )=0

2k+1+k^{2}+1-2+2k =0

k^{2}+4k=0

We have two values. 


Option 1)

exactly three values

This option is incorrect

Option 2)

any value

This option is incorrect

Option 3)

exactly one values

This option is incorrect

Option 4)

exactly two values

This option is correct

option no 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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