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Question #40850

Difficulty: 8/10 | 🟢 Free

The Question

Let $P(1, -1, 0)$ and $Q(1, 1, 2)$ be two points in $\mathbb{R}^3$ . A point $R$ divides the line segment $PQ$ internally in the ratio $k:1$ . Let $A, B,$ and $C$ be the projections of $R$ onto the $xy$ , $yz$ , and $zx$ planes respectively. If the area of $\triangle ABC$ is $\frac{3}{8}$ , then the distance of $R$ from the origin is:

Options

A.
$\frac{\sqrt{6}}{2}$
B.
$\frac{\sqrt{3}}{2}$
C.
$\frac{3}{2}$
D.
$\sqrt{2}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Projection vs. Coordinate|Ratio of Division (k:1)|Even Multiplicity Roots

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle 3D projections and the resulting polynomial equations for the division ratio 'k'.

The area of the projection triangle leads to a quartic equation in 'k'.

The primary trap lies in the 'Even Multiplicity Roots' and the potential for extraneous solutions when taking the square root of the area squared.

Students often expand the quartic blindly, leading to algebraic errors, or fail to recognize the perfect square $(3k^2+1)^2$ , getting bogged down in a degree-4 polynomial.

Furthermore, the negative case $3k^2+1 = -\frac{3}{4}(k+1)^2$ yields complex roots, which students might miscalculate as real extraneous roots if they make sign errors, a common pitfall in advanced coordinate geometry.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Projection vs. Coordinate
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free