Find the Cartesian and vector equations of the planes passing through the intersection of the planes r → · ( i + 3 j ) + 6 = 0 and r → · ( 3 i - j + 4 k ) = 0 , whose perpendicular distance from the origin is 1 unit .

Given that, the equation of the planes are r → · ( i + 3 j ) + 6 = 0 and r → · ( 3 i - j + 4 k ) = 0 . Also, given that, the required plane is passing through the intersection of the given planes. ∴ Equation of required plane is ( x + 3 y + 6 ) + λ ( 3 x – y + 4 z ) = 0 ⇒ ( 1 + 3 λ ) x + ( 3 – λ ) y + 4 λ z + 6 = 0 Perpendicular distance to the plane from origin is unity. Therefore, 6 1 + 3 λ 2 + 3 - λ 2 + 4 λ 2 = 1 Since, perpendicular distance to a plane a x + b y + c z + d = 0 from a point x 1 , y 1 , z 1 is a x 1 + b y 1 + c z 1 + d a 2 + b 2 + c 2 . ⇒ 36 = 1 + 9 λ 2 + 6 λ + 9 + λ 2 – 6 λ + 16 λ 2 ⇒ λ = ± 1 Equations of required planes are ( x + 3 y + 6 ) + 1 ( 3 x – y + 4 z ) = 0 and ( x + 3 y + 6 ) + - 1 ( 3 x – y + 4 z ) = 0 ⇒ 4 x + 2 y + 4 z + 6 = 0 and – 2 x + 4 y - 4 z + 6 = 0 ⇒ 2 x + y + 2 z + 3 = 0 and x – 2 y – 2 z – 3 = 0 Hence, the required equation of the planes are 2 x + y + 2 z + 3 = 0 and x – 2 y + 2 z – 3 = 0 .

E M B I B E

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 25 - The Plane>EXERCISE>Q 1

HARD

12th West Bengal Board

IMPORTANT

Earn 100

Show that the following four points are coplanar, and find the equation of the plane: $(0, 4, 3), (- 1, - 5, - 3), (- 2, - 2, 1)$ and $(1, 1, - 1)$

Important Questions on The Plane

HARD

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 25 - The Plane>EXERCISE>Q 2

Find the equation of the plane passing through the intersection of the planes

x + 2 y + 3 z - 4 = 0

and

2 x + y - z + 5 = 0

and perpendicular to the plane

5 x + 3 y + 6 z + 8 = 0

.

HARD

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 25 - The Plane>EXERCISE>Q 3

Show that the equation of the plane passing through the intersection of the planes

x + y + z = 1

and

2 x + 3 y + 4 z = 5

and perpendicular to the plane

9 x - 5 y + 6 z + 2 = 0

is

x - z + 2 = 0

.

EASY

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 25 - The Plane>EXERCISE>Q 4

Find the equation of the plane through the intersection of the planes

x + 3 y + 6 = 0

and

3 x - y - 4 z = 0

and whose perpendicular distance from origin is unity.

HARD

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 25 - The Plane>EXERCISE>Q 5

Find the Cartesian and vector equations of the planes passing through the intersection of the planes

\vec{r} \cdot (i + 3 j) + 6 = 0

and

\vec{r} \cdot (3 i - j + 4 k) = 0

, whose perpendicular distance from the origin is

1 unit

.

EASY

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 25 - The Plane>EXERCISE>Q 6

Find the equation of the plane passing through the point $A (- 1, - 1, 2)$ and perpendicular to each of the planes $3 x + 2 y - 3 z = 1$ and $5 x - 4 y + z = 5 .$