Let P be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes $\vec{r} \cdot(\hat{i}+\hat{j}+4 \hat{k})=16$ and $\vec{r} \cdot(-\hat{i}+\hat{j}+\hat{k})=6$. Then which of the following points does NOT lie on P ?

JEE Main · 2021 · 26 Aug 2021 · Shift 2 · Three Dimensional Geometry

Let P be the plane passing through the point

(1, 2, 3)

and the line of intersection of the planes

r \cdot (\hat{i} + \hat{j} + 4 \hat{k}) = 16

and

r \cdot (- \hat{i} + \hat{j} + \hat{k}) = 6

.
Then which of the following points does NOT lie on P ?

Answer: (D)

A plane through the line of intersection of the two given planes has the form

(x + y + 4 z - 16) + λ (- x + y + z - 6) = 0

Putting

(1, 2, 3)

into it gives

(1 + 2 + 12 - 16) + λ (- 1 + 2 + 3 - 6) = 0

, that is

- 1 - 2 λ = 0

, so

λ = - \frac{1}{2}

. Multiplying through by

2

2 (x + y + 4 z - 16) - (- x + y + z - 6) = 0 ⟹ 3 x + y + 7 z = 26

Testing the four points in

3 x + y + 7 z

(3, 3, 2)

gives

9 + 3 + 14 = 26

(6, - 6, 2)

gives

18 - 6 + 14 = 26

(- 8, 8, 6)

gives

- 24 + 8 + 42 = 26

, while

(4, 2, 2)

gives

12 + 2 + 14 = 28

. The point that does not lie on

P

(4, 2, 2)

JEE Main · 2021 · 26 Aug 2021 · Shift 2 · Single correct option