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Question #40783

Difficulty: 8/10 | 🟢 Free

The Question

For a parameter $\alpha \in [0, 2\pi)$ , a family of lines is given by $x \cos \alpha + y \sin \alpha = k \cos 2\alpha$ , where $k > 0$ is a constant. Let $C_k$ be the envelope of this family of lines. As $k$ varies over all positive real numbers, a family of curves $\mathcal{F} = \{C_k : k > 0\}$ is generated. Which of the following represents the orthogonal trajectories of the family $\mathcal{F}$ ?

Options

A.
$(x-y)^{4/3} - (x+y)^{4/3} = c$
B.
$(x-y)^{4/3} + (x+y)^{4/3} = c$
C.
$(x-y)^{2/3} - (x+y)^{2/3} = c$
D.
$(x+y)^{4/3} + (x-y)^{2/3} = c$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Normal Form Parameters|Geometric Interpretation|Orthogonal Trajectories

Why This Question Is Tricky

Common Mistakes

This problem tests the geometric interpretation of the normal form and the envelope of a family of lines, combined with orthogonal trajectories.

The primary trap lies in the algebraic complexity of the envelope; students who fail to recognize the symmetry and apply a $\pi/4$ coordinate rotation will face an unintegrable differential equation in the standard $xy$ -plane.

Furthermore, a common conceptual error is attempting to find the orthogonal trajectories of the lines themselves for a fixed $k$ , rather than the envelope family $\mathcal{F}$ generated by varying $k$ , which leads to incorrect slope substitutions and sign errors.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Normal Form Parameters
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free