JEE Mathematics/ Lakshmikanta Satapathy/ 3D Geometry QA part 9/ Advanced Question on Line and Plane taken from previous JEE solved with the related concepts

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L K Satapathy
3D Geometry QA 9

2. Physics Helpline
L K Satapathy
3 D Geometry QA 9
Question: Let P be the point of intersection of the straight line joining the points
Answer :
Q (2,3,5) and R (1 , –1 , 4) with the plane 5x – 4y – z = 1 . If S be the
foot of the perpendicular dropped from the point T (2,1,4) on QR , then
the length of the line segment PS is
1( ) ( ) 2 ( ) 2 ( ) 2 2
2
a b c d
Equation of QR is 11 4
2 1 3 1 5 4
yx z  
  
11 4 . . . (1)
1 4 1
yx z 
    
Any point on QR ( 1,4 1, 4)     
If it lies on plane, then 5( 1) 4(4 1) 1( 4) 1       
112 5 1
3
      
 4 1 13( 1,4 1, 4) , ,
3 3 3
P        
(2,3,5)Q
(1, 1,4)R 
P
S
(2,1,4)T

3. Physics Helpline
L K Satapathy
3 D Geometry QA 9
(2,3,5)Q
(1, 1,4)R 
P
S
(2,1,4)T
For some value of  , coordinates of S ( 1,4 1, 4)     
DRs of TS ( 1 2,4 1 1, 4 4) ( 1,4 2, )              
And DRs of QR (1,4,1)
PR  TS 1( 1) 4(4 2) 1( ) 0       
118 9 0
2
     
 Coordinates of S  3 9( 1,4 1, 4) , 1,
2 2
      
     
2 2 2
4 3 1 13 91
3 2 3 3 2
PS      
1 4 1 1 16 1 1
36 9 36 36 2
[ ]Ans     
Correct option = (a)

4. Physics Helpline
L K Satapathy
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