Question

The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y, z = 1 also passes through the point ______.

Options

• (0, –6, 2)

• (0, 6, –2)

• (–2, 0, 1)

• (2, 0, –1)

Answer 1

The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y, z = 1 also passes through the point (–2, 0, 1).

Explanation:

Plane passing through (1, 2, 1)

a(x – 1) + b(y – 2) + c(z – 1) = 0

Also passing through (2, 1, 2)

a(2 – 1) + b(1 – 2) + c(2 – 1) = 0

a – b + c = 0  ...(i)

And this plane parallel to line

2x = 3y, z = 1

`x/3 = y/2 = (z - 1)/0`

Direction numbers of ⊥ plane all a, b, c and line direction numbers 3, 2, 0

⇒ 3a + 2b + 0c = 0  ...(ii)

From equation no. (i) and (ii)

a – b + c = 0

3a + 2b + 0c = 0

`a/(0 - 2) = (+b)/(3 - 0) = c/(2 + 3)` = λ (let)

a = –2λ, b = +3λ, c = 5λ

Substituting in the equation of plane

–2λ(x – 1) + 3λ(y – 2) + 5λ(z – 1) = 0

2x – 3y – 5z + 9 = 0