The length of the perpendicular drawn from the point $(2,1,4)$ to the plane containing the lines $\vec{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+2 \hat{j}-\hat{k})$ and $\vec{r}=(\hat{i}+\hat{j})+\mu(-\hat{i}+\hat{j}-2 \hat{k})$ is:

JEE Main · 2019 · 12 Apr 2019 · Shift 2 · Three Dimensional Geometry

The length of the perpendicular drawn from the point

(2, 1, 4)

to the plane containing the lines

r = (\hat{i} + \hat{j}) + λ (\hat{i} + 2 \hat{j} - \hat{k})

and

r = (\hat{i} + \hat{j}) + μ (- \hat{i} + \hat{j} - 2 \hat{k})

is:

(A) $\frac{1}{3}$
(B) $3$ ✓ Correct
(C) $\frac{1}{3}$
(D)3

Answer: (B)

Solution

Both lines start at

(1, 1, 0)

, so the plane through them has normal

n = (\hat{i} + 2 \hat{j} - \hat{k}) \times (- \hat{i} + \hat{j} - 2 \hat{k}) = - 3 \hat{i} + 3 \hat{j} + 3 \hat{k}

, i.e.

- x + y + z = 0

. The distance of

(2, 1, 4)

from this plane is

\frac{∣ - 2 + 1 + 4∣}{1 + 1 + 1} = \frac{3}{3} = 3

JEE Main · 2019 · 12 Apr 2019 · Shift 2 · Single correct option

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