SHOW THAT THE POINTS (0,-1,-1),(4,5,1),(3,9,4) AND (-4,4,4) ARE COPLANAR.ALSO FIND THE EQUATION OF THE LINE CONTAINING THEM.

The given point are (0, – 1, – 1), (4, 5, 1), (3, 9, 4) and (– 4, 4, 4).

The equations of plane passing through (0, – 1, – 1) is a(x – 0) + b(y + 1) + c(z + 1) = 0        ...(1)

If (4, 5, 1) lies on (1), then

∴ 4a + 6b + 2c = 0

⇒ 2a + 3b + c = 0      ...(2)

If (3, 9, 4) lies on (1), then

∴ 3a + 10b + 5c = 0    ...(3)

Solving (2) and (3), we have

⇒ a = 5λ, b = – 7λ and c = 11 λ

Substituting the values of a, b and c in (1) we get

Putting (– 4, 4, 4) in (4), we get

5 × (– 4) – 7 × 4 + 11 × 4 + 4 = – 20 – 28 + 44 + 4 = 0

∴ (– 4, 4, 4) lies on (4).

Thus, (0, – 1, – 1), (4, 5, 1), (3, 9, 4) and (– 4, 4, 4) are coplanar.

The equation of the plane containing the given points is 5x – 7y + 11z + 4 = 0.