RAJASTHAN ­ PET Rajasthan PET Solved Paper-2007

  • question_answer The distance between the line\[\overrightarrow{a}=2\hat{i}-2\hat{j}+3\hat{k}\] \[\lambda (\hat{i}+5\hat{j}+\hat{k})=5\]and the plane\[\overrightarrow{r}.(\hat{i}+5\hat{j}+\hat{k})=5\]is

    A)  \[\frac{10}{9}\]

    B)  \[\frac{10}{3\sqrt{3}}\]

    C)  \[\frac{3}{10}\]

    D)  \[\frac{10}{3}\]

    Correct Answer: C

    Solution :

     Here, \[(\hat{i}-\hat{j}+4\hat{k}).(\hat{i}+5\hat{j}+\hat{k})=0\] So, the equation of the line parallel to the plane is The  general  point  on this  line  is\[(\lambda +2,-\lambda -2,4\lambda +3)\]for\[\lambda =0\]the point on the line is  and distance from \[\overrightarrow{r}.(\hat{i}+5\hat{j}+\hat{k})=5\]or\[x+5y+z=5\] \[\therefore \] \[d=\left| \frac{2+5(-2)+3-5}{\sqrt{1+25+1}} \right|\] \[\Rightarrow \] \[d=\left| \frac{-10}{3\sqrt{3}} \right|=\frac{10}{3\sqrt{3}}\]

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