Let a plane \(P\) contain two lines \(\vec{r}=\hat{i}+\lambda(\hat{i}+\hat{j}), \lambda \in R\) and \(\vec{r}=-\hat{j}+\mu(\hat{j}-\hat{k}), \mu \in R\). If \(Q(\alpha, \beta, \gamma)\) is the foot of the perpendicular drawn from the point \(M(1,0,1)\) to \(P\), then \(3(\alpha+\beta+\gamma)\) equals ______.

JEE Main · 2020 · 03 Sep 2020 · Shift 2 · Three Dimensional Geometry

Let a plane

P

contain two lines

r = \hat{i} + λ (\hat{i} + \hat{j}), λ \in R

and

r = - \hat{j} + μ (\hat{j} - \hat{k}), μ \in R

. If

Q (α, β, γ)

is the foot of the perpendicular drawn from the point

M (1, 0, 1)

P

, then

3 (α + β + γ)

equals ______.

Answer: 5

The plane contains both given lines, so its normal is perpendicular to both direction vectors,

(1, 1, 0)

and

(0, 1, - 1)

. Their cross product is

(- 1, 1, 1)

, and the plane passes through

(1, 0, 0)

on the first line, giving

- (x - 1) + y + z = 0

, that is

x - y - z - 1 = 0

. To drop the perpendicular from

M (1, 0, 1)

, evaluate

x - y - z - 1

M

to get

- 1

; stepping along the normal

(1, - 1, - 1)

, whose squared length is

3

, the foot is

Q = (1, 0, 1) + \frac{1}{3} (1, - 1, - 1) = (\frac{4}{3}, - \frac{1}{3}, \frac{2}{3})

. Then

α + β + γ = \frac{4}{3} - \frac{1}{3} + \frac{2}{3} = \frac{5}{3}

, so

3 (α + β + γ) = 5

JEE Main · 2020 · 03 Sep 2020 · Shift 2 · Integer answer