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Need explanation for: - Three Dimensional Geometry - JEE Main

 If the line, \frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3} lies in


the plane,lx+my-z=9,then l^{2}+m^{2}  is equal to:

  • Option 1)

    26

  • Option 2)

    18

  • Option 3)

    5

  • Option 4)

    2

 
Answers (1)
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As we learnt in 

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

 

 line \frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3}=k

lies in plane lx+my-z=9

So (3,-2,-4) statisfies lx+my-z=9

3l-2m+4=9

3l-2m=5                ----------(i)

and (2\hat{i}-\hat{j}+3\hat{k}).(l\hat{i}+m\hat{j}-\hat{k})=0

2l-m=3            -------------(ii)

l=1, m=-1

l^{2}+m^{2}=2


Option 1)

26

This option is incorrect

Option 2)

18

This option is incorrect

Option 3)

5

This option is incorrect

Option 4)

2

This option is correct

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