Let the lines $\frac{x-1}{\lambda}=\frac{y-2}{1}=\frac{z-3}{2}$ and $\frac{\mathrm{x}+26}{-2}=\frac{\mathrm{y}+18}{3}=\frac{\mathrm{z}+28}{\lambda}$ be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lies on P?

JEE Main · 2022 · 28 Jul 2022 · Shift 2 · Three Dimensional Geometry

Let the lines

\frac{x - 1}{λ} = \frac{y - 2}{1} = \frac{z - 3}{2}

and

\frac{x + 26}{- 2} = \frac{y + 18}{3} = \frac{z + 28}{λ}

be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lies on P?

Answer: (D)

For the lines to be coplanar, the determinant formed by the vector joining points on the two lines and their direction ratios must be zero:

27 λ - 2 2013 312 λ = 0

. Expanding gives

λ = 3

. The normal to the plane containing both lines is

n = \hat{i} 3 - 2 \hat{j} 13 \hat{k} 23 = - 3 \hat{i} - 13 \hat{j} + 11 \hat{k}

. Using point

(1, 2, 3)

on the first line, the plane equation is

3 x + 13 y - 11 z + 4 = 0

. Checking option (D):

3 (0) + 13 (4) - 11 (5) + 4 = 52 - 55 + 4 = 1 \neq = 0

, so

(0, 4, 5)

does not lie on P.

JEE Main · 2022 · 28 Jul 2022 · Shift 2 · Single correct option