If the product of distances of the point (1, 1, 1) from the origin and the plane x – y + z + λ = 0 be 5, find the value of λ.
The distance of the point (1, 1, 1) from the origin
We know, distance of (x1, y1, z1) form the origin is :
Putting values of x1, y1, z1= 1
⟹ Required Distance = √3
Distance of the point (1, 1, 1) from plane x – y + z + λ = 0
We know, the distance of point (x1,y1,z1) from the plane
is given by:
Putting the necessary values,
⟹
⟹
According to question, the product of the above two distances is 5
⟹
⟹ |1 + λ| = 5
⟹ 1 + λ = 5 or 1 + λ = –5
⟹ λ = 4 or λ = –6