Question

The plane E ≡ ax + by + cz + d = 0 contains the line

L : (x - x₁)/l = (y - y₁)/m = (z - z₁)/n

if and only if (a) al + bm + cn = 0 and (b) ax₁ + by₁ + cz₁ + d = 0

Answer 1

Suppose the line L is contained in the given plane. Since (x₁, y₁, z₁) lies on L and L is contained in E = 0 we have 

ax₁ + by₁ + cz₁ + d = 0

Also the normal (a,b,c) of E = 0 is perpendicular to the line L. This implies 

(a,b,c).(l,m,n) = 0

al + bm + cn = 0 

Conversely, assume that

ax₁ + by₁ + cz₁ + d = 0 ...(1)

and al + bm + cn = 0 ...(2)

Since one of a, b and c is not zero, we have that the vector (a, b, c) is normal to the plane E ≡ ax + by + cz + d = 0. Also from Eqs. (1) and (2), we have that (x₁, y₁, z₁) lies in E = 0 and al + bm + cn = 0 which implies that (l,m,n) is perpendicular to the normal (a,b,c) or E = 0. Hence, L must lie in the plane E = 0.