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Question #40545

Difficulty: 8/10 | 🟢 Free

The Question

Let $R$ be the set of points $(x, y)$ in the Cartesian plane satisfying the following conditions: $\frac{x - y}{x + y} \ge 1, \quad \frac{x + y}{x - y} \le -1, \quad \text{and} \quad x^2 + y^2 \le 16$ What is the area of the region $R$ ?

Options

A.
$0$
B.
$4\pi$
C.
$8\pi$
D.
$16\pi$

Check Your Answer

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

Cross Multiplication Trap|Unbounded Region Trap|Inconsistency Detection

Why This Question Is Tricky

Common Mistakes

This question tests the 'Cross-Multiplication Trap' and 'Inconsistency Detection'.

Students often blindly cross-multiply inequalities with variables in the denominator, incorrectly yielding $x - y \ge x + y \implies y \le 0$ and $x + y \le -x + y \implies x \le 0$ .

This false region (the third quadrant) bounded by the circle gives an area of $4\pi$ .

Rigorous algebraic manipulation using homogeneous variables ( $m = y/x$ ) reveals the true feasible region is the null set, turning a standard area problem into an existence proof.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Cross Multiplication Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free