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  • Try this! - Let S be the set of all real values ofsuch that a plane passing through the points also passes t - Three Dimensional Geometry - JEE Main-11370

Let S be the set of all real values of \lambda such that a plane passing through the points \left ( -\lambda ^{2},1,1 \right ),\left ( 1,-\lambda ^{2},1 \right )\: and\: \left ( 1,1,-\lambda ^{2} \right )  also passes through the point \left ( -1,-1,1 \right ). Then S is equal to : 

  • Option 1)

     

    \left \{ \sqrt{3} \right \}
     

     

     

     

  • Option 2)

     

    \left \{ 3,-3 \right \}

  • Option 3)

     

    \left \{ 1,-1 \right \}

  • Option 4)

     

    \left \{ \sqrt{3},-\sqrt{3}\right \}

Answers (1)

best_answer

 

Plane passing through three points (cartesian form) -

Let the plane passes through

A(x_{1},y_{1},z_{1}),B(x_{2},y_{2},z_{2})\: and \: C(x_{3},y_{3},z_{3})

then the plane is given by

\begin{vmatrix} x-x_{1} &y-y_{1} &z-z_{1} \\ x-x_{2} & y-y_{2} & z-z_{2}\\ x-x_{3}&y-y_{3} & z-z_{3} \end{vmatrix}=0
 

 

- wherein

\underset{AB}{\rightarrow} = \left ( x_{2}-x_{1} \right )\hat{i}+ \left ( y_{2}-y_{1} \right )\hat{j}+ \left ( z_{2}-z_{1} \right )\hat{k}

\underset{AC}{\rightarrow} = \left ( x_{3}-x_{1} \right )\hat{i}+ \left ( y_{3}-y_{1} \right )\hat{j}+ \left ( z_{3}-z_{1} \right )\hat{k}

\vec{n}=\underset{AB}{\rightarrow} \times\underset{AC}{\rightarrow}

\left ( \vec{r} -\vec{a}\right )\cdot\underset{AB}{\rightarrow} \times\underset{AC}{\rightarrow}= 0

 

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

 

-(\lambda ^{2}+1)\left \{ \left ( 1-\lambda ^{2} \right )^{2}-4 \right \}=0\\\\\Rightarrow \lambda ^{2}-1=\pm 2\\\\\Rightarrow \lambda =\pm \sqrt{3}


Option 1)

 

\left \{ \sqrt{3} \right \}
 

 

 

 

Option 2)

 

\left \{ 3,-3 \right \}

Option 3)

 

\left \{ 1,-1 \right \}

Option 4)

 

\left \{ \sqrt{3},-\sqrt{3}\right \}
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