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Question #41628

Difficulty: 8/10 | 🟢 Free

The Question

Let $P$ be the point of intersection of the line $L: \frac{x-1}{2} = \frac{y+1}{-1} = \frac{z-2}{2}$ and the plane $\Pi_1: x + y + z = 5$ . Let $Q$ be the image of $P$ in the plane $\Pi_2: x - 2y + 2z = 6$ . If $d$ is the length of the projection of the vector $\vec{PQ}$ on the line $L$ , and $\theta$ is the acute angle between the line $L$ and the plane $\Pi_2$ , find the value of $9d^2 + 81 \cos^2\theta$ .

Options

A.
273
B.
320
C.
2321
D.
81

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Image of a point in a plane|Angle between line and plane|Parametric form pitfalls

Why This Question Is Tricky

Common Mistakes

This question tests two major coordinate geometry traps.

First, the 'Parametric Form Pitfall': students often fail to normalize the direction vector $\vec{b}$ when calculating the projection length $d$ , erroneously taking the dot product directly and yielding $d=16$ instead of $16/3$ , leading to the distractor 2321.

Second, the 'Angle Trap': students frequently confuse the angle between a line and a plane ( $\sin\theta$ ) with the angle between two vectors ( $\cos\theta$ ), incorrectly computing $\cos\theta = \frac{\vec{b} \cdot \vec{n}}{|\vec{b}||\vec{n}|} = \frac{8}{9}$ , which leads to the distractor option 320.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Image of a point in a plane
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free