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Question #40694

Difficulty: 8/10 | 🟢 Free

The Question

Let $t_1, t_2, t_3$ be three distinct positive real numbers in a geometric progression such that $t_1 t_2 t_3 = 8$ . A moving point $P(x, y)$ is defined by its coordinates $x = t_1 + t_2 + t_3$ and $y = t_1 t_2 + t_2 t_3 + t_3 t_1$ . If the locus of $P$ intersects the region $(x - 3)^2 + (y - 4)^2 \le R^2$ , then the infimum of $R^2$ is

Options

A.
73
B.
$\frac{4}{5}$
C.
25
D.
85

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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

Selection of GP Terms|Arithmetic Mean Inequality|Geometry Constraint Modeling|Locus of a Moving Point

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of sequence optimization (AM-GM) with coordinate geometry constraints.

The primary trap lies in decoupling the algebraic locus from its domain restrictions.

Students easily identify the linear relationship $y = 2x$ and proceed to minimize the distance from the point $(3, 4)$ to this line, yielding the extraneous minimum distance squared of $4/5$ (Trap 109).

However, applying the AM-GM inequality to the distinct positive GP terms reveals a strict boundary condition: $x > 6$ (Trap 269).

The true infimum must be evaluated at this boundary, forcing students to backtrack and ensure the geometric intersection is valid within the algebraically permissible domain of the GP.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Selection of GP Terms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free