If for some $\alpha \in R$, the lines $L_{1}: \frac{x+1}{2}=\frac{y-2}{-1}=\frac{z-1}{1}$ and $L_{2}: \frac{x+2}{\alpha}=\frac{y+1}{5-\alpha}=\frac{z+1}{1}$ are coplanar, then the line $L_{2}$ passes through the point:

JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Three Dimensional Geometry

If for some

α \in R

, the lines

L_{1} : \frac{x + 1}{2} = \frac{y - 2}{- 1} = \frac{z - 1}{1}

and

L_{2} : \frac{x + 2}{α} = \frac{y + 1}{5 - α} = \frac{z + 1}{1}

are coplanar, then the line

L_{2}

passes through the point:

Answer: (A)

The two lines are coplanar when the determinant built from the joining vector and the two direction vectors vanishes,

α 21 5 - α - 1 3 112 = 0

. Expanding gives

- 2 α - 8 = 0

, so

α = - 4

, and the second line becomes

\frac{x + 2}{- 4} = \frac{y + 1}{9} = \frac{z + 1}{1}

. Testing

(2, - 10, - 2)

, every ratio equals

\frac{2 + 2}{- 4} = \frac{- 10 + 1}{9} = \frac{- 2 + 1}{1} = - 1

, so the line passes through

(2, - 10, - 2)

JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Single correct option