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Help me solve this - Three Dimensional Geometry - JEE Main

 The number of distinct real values of λ for which the lines

\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{\lambda ^{2}} and

\frac{x-3}{1}=\frac{y-2}{\lambda ^{2}}=\frac{z-1}{2}

are coplanar is :

  • Option 1)

    4

  • Option 2)

    1

  • Option 3)

    2

  • Option 4)

    3

 
Answers (1)
98 Views

As we learnt in 

Condition for lines to be intersecting (cartesian form) -

Their shortest distance should be 0

Also the condition for coplanar lines

-

 

For two lines to be coplanar

[a^{-1}-b^{-1}\, \, \, \, \, \, r_{1}^{-1}\, \, \, \, \, r_{2}^{-1}]=0

\begin{vmatrix} 2 & 0&4 \\ 1 &2 &\lambda^{2} \\ 1 & \lambda^{2} & 2 \end{vmatrix}=0

2(4-\lambda^{4})+4(\lambda^{2}-2)=0

\lambda^{4}=2\lambda^{2}

\lambda^{4}-2\lambda^{2}=0

\lambda^{2}(\lambda^{2}-2)=0

But \lambda\pm 0\: \therefore \lambda=\pm \sqrt{2}

Two values of \lambda are possible


Option 1)

4

This option is incorrect

Option 2)

1

This option is incorrect

Option 3)

2

This option is correct

Option 4)

3

This option is incorrect

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