Find the equation of a plane (in a x + b y + c z = d form) which is at a distance of 3 3 units from the origin and the normal to which is equally inclined to the coordinate axes.

Given that the normal to the plane is equally inclined to the coordinate axis. Now if α , β , and γ are the angles made by the line segment with the coordinate axis. Then, we have cos α = cos β = cos γ We know that cos 2 α + cos 2 β + cos 2 γ = 1 ⇒ 3 cos 2 α = 1 ⇒ cos α = 1 3 ∵ α is an acute angle So, the normal to the plane is N → = 1 3 i ^ + 1 3 j ^ + 1 3 k ^ and plane is at a distance of 3 3 units from origin. We know that the equation of the plane at a distance p from the origin where N ^ is the unit vector normal to the plane is r → · N ^ = p . . . . . 1 Now, N → = 1 ( 3 ) 2 + 1 ( 3 ) 2 + 1 ( 3 ) 2 = 3 3 = 1 ⇒ N ^ = N → N → = 1 3 i ^ + 1 3 ȷ ^ + 1 3 k ^ 1 = 1 3 i ^ + 1 3 ȷ ^ + 1 3 k ^ Therefore, the equation of the required plane using 1 is ⇒ x i ^ + y j ^ + z k ^ · 1 3 i ^ + 1 3 ȷ ^ + 1 3 k ^ 1 = 3 3 ⇒ x 3 + y 3 + z 3 = 3 3 ⇒ x + y + z = 3 3 · 3 = 9 So, the required equation of plane is x + y + z = 9 .

E M B I B E

MATHEMATICS CLASS XII VOLUME-2>Chapter 10 - The Plane>EXERCISE>Q 1

EASY

12th CBSE

IMPORTANT

Earn 100

Find the vector equation of a plane which is at a distance of $3$ units from the origin and has $\hat{k}$ as the unit vector normal to it.

Important Questions on The Plane

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 10 - The Plane>EXERCISE>Q 2

Find the vector equation of a plane which is at a distance of

5

units from the origin and which is normal to the vector

\hat{i} - 2 \hat{j} - 2 \hat{k}

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 10 - The Plane>EXERCISE>Q 3

Reduce the equation

2 x - 3 y - 6 z = 14

to the normal form and, hence, find the length of the perpendicular from the origin to the plane. Also, find the direction cosines of the normal to the plane.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 10 - The Plane>EXERCISE>Q 4

Reduce the equation