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Question #40813

Difficulty: 8/10 | 🟢 Free

The Question

Let $A, B, C$ be the angles of an acute-angled triangle. A point $P$ is defined as $P(\sin^2 A + \sin^2 B + \sin^2 C, \cos^2 A + \cos^2 B + \cos^2 C)$ . Let $L$ be the length of the chord of contact of the tangents drawn from $P$ to the circle $x^2 + y^2 = \frac{9}{2}$ . Find the maximum possible value of $L$ .

Options

A.
$\frac{3\sqrt{10}}{5}$
B.
3
C.
$\frac{3\sqrt{5}}{5}$
D.
$\frac{3\sqrt{2}}{2}$

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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

Chord of Contact Logic|Conditional Identity Constraints|Monotonicity and Boundedness

Why This Question Is Tricky

Common Mistakes

This question tests the integration of conditional trigonometric identities with coordinate geometry, specifically targeting the 'Monotonicity and Boundedness' trap.

Students often blindly apply derivative-based optimization to the distance function $D^2(x) = 2x^2 - 6x + 9$ without establishing the strict domain $x \in (2, 9/4]$ derived from the acute triangle constraint $\sum \sin^2 A = 2 + 2\cos A\cos B\cos C$ .

Failing to recognize this bounded domain leads to incorrect boundary evaluations: assuming $x \to 0$ yields a false maximum chord length of 3, while using the unconstrained vertex $x = 1.5$ places the point on the circle, resulting in a chord length of 0.

The true maximum lies at the boundary $x = 9/4$ , requiring careful verification of the domain and monotonicity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Chord of Contact Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free