3D Geometry
Perpendicularity condition for two lines in space
MMTS_Full_Test_11
Grade 12
Question:
Consider lines $L_1:\{3\sqrt3x=4\sin\theta\,y+(2\sqrt{\sin\theta}-3)\,3\sqrt3,\;3\sqrt3z=(2\sqrt{\sin\theta}-3)y+3\sqrt3\sin\theta\}$ and $L_2:\{\sqrt3x=-2\sqrt{\cos\theta}\,y+\sqrt3(3-2\sqrt{\cos\theta}),\;\sqrt3z=(3-2\sqrt{\cos\theta})y+\sqrt3\cos\theta\}$. If $L_1\perp L_2$ then
(A) $\theta=n\pi+\dfrac{\pi}{4}$; $n\in I$
(B) $\theta=n\pi+\dfrac{\pi}{4}$; $n$ is even
(C) $\theta=n\pi+\dfrac{\pi}{4}$; $n$ is odd
(D) $\theta=n\pi$; $n\in I$
Step-by-Step Solution
Key Concept: Extract direction ratios of $L_1$ and $L_2$ in terms of $\theta$. Set dot product to zero: $\sin\theta\cos\theta-\sin\theta-\cos\theta+\ldots=0$.
$\theta=n\pi+\pi/4$ with $n$ odd.
Correct Answer: (C) $\theta=n\pi+\dfrac{\pi}{4}$; $n$ is odd