Find the equation of plane passing through (3,2,0)and contain the line (x-3/1,y-6/5,z-4/4)

Question

Arun · Answer

L e t t h e e q u a t i o n o f a p l a n e p a s sin g t h r o u g h ( 3 , 2 , 0 ) b e a ( x − 3 ) + b ( y − 2 ) + c ( z − 0 ) = 0 . . ( 1 ) T h e l i n e x − 3 1 = y − 6 5 = z − 4 4 p a s s e s t h r o u g h t h e p o i n t ( 3 , 2 , 0 ) a n d h a s d i r e c t i o n r a t i o s 1 , 5 , 4 . I f ( 1 ) c o n t a i n s t h e l i n e s , i t m u s t p a s s t h r o u g h ( 3 , 6 , 4 ) a n d m u s t b e p a r a l l e l t o t h e l i n e . T h e r e f o r e , a ( 3 − 3 ) + b ( 6 − 2 ) + c ( 4 − 0 ) = 0 ⇒ 4 b + 4 c = 0 ⇒ b + c = 0 . . . ( 2 ) a n d a + 5 b + 4 c = 0 . . . ( 3 ) O n s o l v i n g ( 2 ) a n d ( 3 ) , w e g e t a = c a n d b = − c S o , F r o m e q ( 1 ) , w e g e t c ( x − 3 ) − c ( y − 2 ) + c ( z − 0 ) = 0 ⇒ c [ ( x − 3 ) − ( y − 2 ) + z ] = 0 ⇒ ( x − 3 ) − ( y − 2 ) + z = 0 ⇒ x − y + z − 1 = 0 ⇒ x − y + z = 1 i s t h e r e q u i r e d e q u a t i o n o f p l a n e .

Arun · Answer

Let the equation of plane passing through (3,2,0) is a(x-3) + b(y-2) + c(z-0) = 0 x-3/1 = y-6/5 = z-4/4 Pass through (3,2,0) and has direction ratios as 1,5,4 If the equation of plane contains this, it should pass from (3,6,4) and parallel to (1,5,4) Hence a(3-3) + b(6-2) + c *4 = 0 4b + 4c = 0 b +c = 0 And a + 5b +4 c = 0 On solving these two equations a = c & b= -c Hence equation of plane will be c(x-3) - c(y-2) +cz = 0 x-3 -y +2 +z= 0 x - y +z -1 = 0 x -y +z = 1

Rishi Sharma · Answer

Dear Student,
Please find below the solution to your problem.

Let the equation of plane passing through (3,2,0) is
a(x-3) + b(y-2) + c(z-0) = 0
x-3/1 = y-6/5 = z-4/4
Pass through (3,2,0) and has direction ratios as 1,5,4
If the equation of plane contains this, it should pass from (3,6,4) and parallel to (1,5,4)
Hence
a(3-3) + b(6-2) + c *4 = 0
4b + 4c = 0
b +c = 0
And
a + 5b +4 c = 0
On solving these two equations
a = c & b= -c
Hence equation of plane will be
c(x-3) - c(y-2) +cz = 0
x-3 -y +2 +z= 0
x - y +z -1 = 0
x -y +z = 1

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Find the equation of plane passing through (3,2,0)and contain the line (x-3/1,y-6/5,z-4/4)

Find the equation of plane passing through (3,2,0)and contain the line (x-3/1,y-6/5,z-4/4)

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