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  • How to solve this problem- - Three Dimensional Geometry - JEE Main-3

If the shortest distance between the lines x+2\lambda =2y=-12z,x=y+4\lambda =6z-12\lambda \, \, is\, 4\sqrt{2}\; units,

then   a value of \lambda is:

  • Option 1)

    \frac{\sqrt{2}}{2}

  • Option 2)

    2

  • Option 3)

    \sqrt{2}

  • Option 4)

    2\sqrt{2}

 

Answers (1)

best_answer

 

Shortest distance between two skew lines (Cartesian form) -

Shortest distance between

\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}}is given by

\left | \frac{\left ( \vec{b} \times \vec{b_{1}}\right )\cdot \left ( \vec{a} -\vec{a_{1}}\right )}{\left | \vec{b} \times \vec{b_{1}} \right |} \right | Where 

\vec{a}= x_{1}\hat{i}+y_{1}\hat{j}+z_{1}\hat{k}

\vec{a_{1}}= x_{2}\hat{i}+y_{2}\hat{j}+z_{2}\hat{k}

\vec{b}= a_{1}\hat{i}+b_{1}\hat{j}+c_{1}\hat{k}

\vec{b_{1}}= a_{2}\hat{i}+b_{2}\hat{j}+c_{2}\hat{k}

 

-

 

 \frac{x+2\lambda }{-12}=\frac{y}{-6}=\frac{z}{1}  & \frac{x}{6}=\frac{y+4\lambda }{6}=z-2\lambda

For non parallel lines,

Distance = \left | \frac{\left [ \left ( \overrightarrow{a_2}-\overrightarrow{a_1} \right )\overrightarrow{b_1}.\overrightarrow{b_2} \right ]}{\overrightarrow{b_1}\times\overrightarrow{b_2}} \right |

We have

\overrightarrow{a_1}= -2\lambda \hat{i}

\overrightarrow{b_1}= -12\hat{i}-6\hat{j}+\hat{k}

\overrightarrow{a_2}= -4\lambda \hat{i}+2\lambda \hat{k}

\overrightarrow{b_2}= 6\hat{i}+6\hat{j}+\hat{k}

Distance = \left | \frac{\left [ \left ( 2\lambda \hat{i}-4\lambda \hat{j}+2\lambda \hat{k} \right )\overrightarrow{b_1}.\overrightarrow{b_2} \right ]}{\overrightarrow{b_1}\times\overrightarrow{b_2}} \right |= 4\sqrt{2}\left ( given \right )

\overrightarrow{b_1}\times \overrightarrow{b_2}= \begin{vmatrix} \hat{i} &\hat{j} & \hat{k}\\ -12 &-6 &1 \\6 &6 &1 \end{vmatrix}= -12\hat{i}+18\hat{j}-36\hat{k}

4\sqrt{2}= \frac{168\lambda }{42}\Rightarrow \lambda = \sqrt{2}

 

 


Option 1)

\frac{\sqrt{2}}{2}

Option 2)

2

Option 3)

\sqrt{2}

Option 4)

2\sqrt{2}

Posted by

satyajeet

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