Given – The equation of the plane is given by ax + by + d = 0

To prove – The plane is parallel to z - axis

Tip – If ax + by + cz + d is the equation of the plane then its angle with the z - axis will be given by

Considering the equation, the direction ratios of its normal is given by (a, b, 0)

The angle the plane makes with the z - axis = sin ^{- 1}[0/√(a² + b²)] = 0

Hence, the plane is parallel to the z - axis

To find – Equation of the plane parallel to z - axis and passing through points A = (2, - 3, 1) and B = (- 4, 7, 6)

The given equation ax + by + d = 0 passes through (2, - 3, 1) & (- 4, 7, 6)

………(i)

……(ii)

Solving (i) and (ii),

[α → arbitrary constant]

Substituting the values of a and b in eqn (i), we get,

- 2Х10α + 3Х6α + d = 0 i.e. d = - 2α

Putting the value of a, b and d in the equation ax + by + d = 0,

(- 10α)x + (- 6α)y + (- 2α) = 0

i.e. 5x + 3y + 1 = 0