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Question #40912

Difficulty: 8/10 | 🟢 Free

The Question

A point $P(x,y)$ moves in the $xy$ -plane such that the sum of the squares of its projections on the coordinate axes is $4$ . If $P$ satisfies the relation $\sin^{-1}\left(\frac{3x^2 y - y^3}{8}\right) = \cos^{-1}\left(\frac{x^3 - 3xy^2}{8}\right)$ then the total length of the locus of $P$ is:

Options

A.
$\pi$
B.
$4\pi$
C.
$\frac{\pi}{3}$
D.
$\frac{2\pi}{3}$

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

Coordinate Axis Projections|Locus of a Moving Point|Algebraic-Trigonometric Graphical Mapping

Why This Question Is Tricky

Common Mistakes

This question tests the integration of coordinate geometry loci with the periodic branch cuts of inverse trigonometric functions.

The trap lies in the 'Quadrant Adjustment Logic'.

Students often incorrectly assume the identity $\sin^{-1}(\sin x) = \cos^{-1}(\cos x) = x$ holds universally, ignoring the principal value branches.

This leads to the spurious conclusion that the equation holds for all $\theta \in [0, 2\pi)$ , yielding the full circumference $4\pi$ .

Another common error is finding only the first-quadrant solution, yielding $\pi/3$ .

The rigorous solution requires mapping the $3\theta$ argument across multiple cycles to find the exact valid arc segments.

Quick Facts About This Question

Question Information

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