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The plane containing the line \frac{x-3}{2}=\frac{y+2}{-1}=\frac{z-1}{3}  and also containing its projection on the plane 2x+3y-z=5 ,  contains which one of the following points?

  • Option 1)

     

    (0,-2,2)

  • Option 2)

     

    (-2,2,2)

  • Option 3)

     

    (2,0,-2)

  • Option 4)

     

    (2,2,0)

Answers (1)

best_answer

 

Plane passing through a point and a line (vector form) -

Let the plane passes through A(\vec{a}) and a line \vec{r}= \vec{b}+\lambda \vec{c}, then the plane is given by

\left [ r\: b\: c \right ]+\left [ r\: c\: a \right ]= \left [a\: b\: c \right ]

 

- wherein

\vec{n}= \left ( \vec{b}- \vec{a} \right )\times \left ( \vec{c} \right )

\left ( \vec{r} -\vec{a}\right )\cdot \left ( \vec{b}-\vec{a} \right )\times\left ( \vec{c} \right )= 0

 

Normal vector of the plane is 

\left ( 2\hat{i}-\hat{j}+3\hat{k} \right )\times \left ( 2\hat{i}+3\hat{j}-\hat{k} \right )=8\left ( -\hat{i}+\hat{j}+\hat{k} \right )

Hence, the equation of plane is 

-x+y+z=\lambda   It satisfies \left ( 3,-2,1 \right )

\Rightarrow Equation of plane -x+y+z=-y


Option 1)

 

(0,-2,2)

Option 2)

 

(-2,2,2)

Option 3)

 

(2,0,-2)

Option 4)

 

(2,2,0)

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