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Question #41149

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all real numbers $r > 0$ such that the curve $y = \sqrt{x^2 - 9}$ and the circle $x^2 + (y+4)^2 = r^2$ intersect at exactly two distinct points. If $m = \inf(S)$ , find the value of $m^2$ .

Options

A.
17
B.
25
C.
34
D.
9

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

Domain Restrictions|Condition of Tangency|Domain and Range Nuance

Why This Question Is Tricky

Common Mistakes

This question tests the 'Domain/Range Nuance' trap in coordinate geometry.

Students typically find the intersection of the curves algebraically or use calculus to equate derivatives, both of which yield a tangency condition at $y = -2$ and $r^2 = 17$ .

However, this algebraic point of tangency violates the range restriction $y \ge 0$ of the principal square root function $y = \sqrt{x^2 - 9}$ .

The true boundary is found by enforcing the range constraint on the resulting quadratic, requiring it to have a non-negative root, which correctly gives $r^2 \ge 25$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Domain Restrictions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free