JEE Mathematics/ Lakshmikanta Satapathy/3D Geometry theory part 9/ Equation of plane in intercept form and plane passing through the line of intersection of two planes

1. Physics Helpline
L K Satapathy
3D Geometry Theory 9

2. Physics Helpline
L K Satapathy
Equation of Plane in intercept form :
The Plane
Consider the plane shown in the figure .
O
B
A X
Z
Y
CLet OA = a , OB = b and OC = c .
 Coordinates of ( ,0,0)A a
 OA , OB and OC are the x , y and z intercepts of the plane.
It cuts the coordinate axes at the points A , B and C.
For a point on the x-axis , its y and z coordinates are both zero
 Coordinates of (0, ,0)B b
 Coordinates of (0,0, )C c
For a point on the y-axis , its x and z coordinates are both zero
For a point on the z-axis , its x and y coordinates are both zero
3 D Geometry Theory 9

3. Physics Helpline
L K Satapathy
The Plane
is a point on it , then (1)  0
D
Aa D A
a
    ( ,0,0)A a
is a point on it , then (1)  0
D
Bb D B
b
    (0, ,0)B b
is a point on it , then (1)  0
D
Cc D C
c
    (0,0, )C c
Consider the general equation of the plane
0 . . . (1)Ax By Cz D   
(1) 0
D D D
x y z D
a b c
      
1 0
x y z
a b c
     
1
x y z
a b c
    [ Cartesian equation ]
3 D Geometry Theory 9

4. Physics Helpline
L K Satapathy
The Plane
Equation of Plane passing through the line of intersection of two planes :
Consider the equation of a plane passing through a point having position vector
 We have
( ). 0r a n 
and perpendicular to a given vector , given by
a
n
. . 0r n a n 
. .r n a n 
Putting we get.a n d .r n d
If ( A , B , C ) be the direction ratios of , thenn ˆˆ ˆn Ai Bj Ck  
3 D Geometry Theory 9

5. Physics Helpline
L K Satapathy
The Plane
Consider two planes and as shown in the figure1 2
Let their equations be
Since L is the line of intersection of the two planes
L 1
2
Position vector of any point on L will satisfy both the equations
Adding equations (1) and (3) , we get 1 2 1 2. .r n r n d d   
1 2 1 2. ( ) . . . (4)r n n d d    
1 1. . . . (1)r n d
2 2. . . . (2)r n d
Rewriting equation (2) , we get 2 2. . . . (3)r n d 
Vector Form :
3 D Geometry Theory 9

6. Physics Helpline
L K Satapathy
The Plane
Now consider a planes passing through L as shown3
 Position vector of any point on it , which satisfies L 1
2
3
r
both (1) and (2) simultaneously , will also satisfy (4)
We observe that a plane passing through the line of
Hence for different values of  , the plane will have
different orientations but it will always pass through L
intersection L , can rotate about the line . This is
indicated by the parameter  .
 Equation of a plane passing through the line of intersection of two planes is
1 2 1 2. ( )r n n d d   
3 D Geometry Theory 9

7. Physics Helpline
L K Satapathy
The Plane
L 1
2
3
Let the direction ratios of the normal to the plane
Cartesian Form :
1
2
1 1 1( , , )A B C
2 2 2 2
ˆˆ ˆand n A i B j C k  
Let the coordinates of any point P on the plane be ( , , )x y z
ˆˆ ˆr xi yj zk   
2 2 2( , , )A B C
and the direction ratios of the normal to the plane
1 1 1 1
ˆˆ ˆn Ai B j C k   
3
3 D Geometry Theory 9

8. Physics Helpline
L K Satapathy
The Plane
[ Cartesian equation ]
Putting the values in equation (4) , we get
1 1 1 2 2 2 1 2
ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ( ). [( ) ( )]xi yj zk Ai B j C k A i B j C k d d         
1 1 1 1 2 2 2 2( ) ( ) 0A x B y C z d A x B y C z d        
1 1 1
ˆ ˆˆ ˆ ˆ ˆ( ). ( )xi yj zk Ai B j C k    
2 2 2 1 2
ˆ ˆˆ ˆ ˆ ˆ( ). ( )xi yj zk A i B j C k d d       
1 1 1 2 2 2 1 2( ) ( )A x B y C z A x B y C z d d        
3 D Geometry Theory 9

9. Physics Helpline
L K Satapathy
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