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Question #41655

Difficulty: 8/10 | 🟢 Free

The Question

Let $L$ be the line given by $x - 1 = 2 - y = \frac{z - 3}{\sqrt{2}}$ . Let $A(1, 2, 3)$ be a point on $L$ , and $B$ be another point on $L$ such that the distance $AB = 2$ . For each positive integer $n$ , let $\Pi_n$ be the plane defined by the equation $\sqrt{n^2+n-1}(x+y) + \sqrt{2}z = n^3$ . Let $p_n$ denote the length of the orthogonal projection of the line segment $AB$ onto the plane $\Pi_n$ . Find the value of $\sum_{n=1}^{99} p_n^2$ .

Options

A.
394.02
B.
395.01
C.
394.00
D.
396.00

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

Orthogonal projection of a line|Parametric Form Manipulation|Method of Differences

Why This Question Is Tricky

Common Mistakes

This question tests the orthogonal projection of a line segment onto a family of planes and the evaluation of the resulting series using the Method of Differences.

The primary trap lies in the parametric manipulation: students often attempt to find the explicit coordinates of the projected points $A'_n$ and $B'_n$ using the line's parametric form, which leads to an intractable algebraic mess.

The optimal path requires recognizing that the projection length depends only on the segment length and the angle between the line's direction vector and the plane's normal.

A secondary trap is the telescoping sum itself; students frequently miss the factor of 2 when squaring the projection length or miscalculate the final boundary terms of the telescoping series, leading to incorrect options like 395.01.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Orthogonal projection of a line
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free