Question

Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 16. Let T be a plane passing through the point Q and contains the line `vecr = -hatk + λ(hati + hatj + 2hatk)`, λ ∈ R. Then, which of the following points lies on T?

Options

• (2, 1, 0)

• (1, 2, 1)

• (1, 2, 2)

• (1, 3, 2)

Answer 1

(1, 2, 1)

Explanation:

Given: Plane: x + 2y + 2z – 16 = 0

And line: `vecr = -hatk + λ(hati + hatj + 2hatk)`

So, equation of line in symmetric form is `(x - 0)/1 = (y - 0)/1 = (z + 1)/2`

Now, mirror image of p(1, 2, 1) in plane x + 2y + 2z – 16 = 0 is

⇒ `(x - 1)/1 = (y - 2)/2 = (z - 1)/2 = -2((1 + 2(2) + 2(1) - 16)/(1^2 + 2^2 + 2^2))`

⇒ `(x - 1)/1 = (y - 2)/2 = (z - 1)/2` = 2

⇒ x = 3, y = 6, z = 5

∴ Coordinates of point Q are (3, 6, 5)

Now, equation of plane T is `|(x, y, z + 1),(1, 1, 2),(3, 6, 6)|` = 0

⇒ x(6 – 12) – y(6 – 6) + (z + 1)(6 – 3) = 0

⇒ –6x + 3z + 3 = 0

⇒ 2x – z – 1 = 0

So, (1, 2, 1) lies on plane T.