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Question #40763

Difficulty: 8/10 | 🟢 Free

The Question

Consider the points $A(0, 5)$ and $B(27, 8)$ in the $xy$ -plane. Let $P$ be a point on the $x$ -axis whose abscissa is given by $x = 2\sec^2 t - 4\sec t + 14$ for some $t \in \left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right]$ . Find the minimum possible value of $PA + PB$ .

Options

A.
30
B.
$\sqrt{898}$
C.
32
D.
$5\sqrt{17} + \sqrt{113}$

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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

Reflection Principle|Triangle Inequality Mind Games|Maximum/Minimum of Quadratic Expressions|Global Extrema on Intervals

Why This Question Is Tricky

Common Mistakes

This question tests the integration of the geometric Reflection Principle with algebraic domain constraints.

The common trap is blindly applying the Reflection Principle to find the straight-line distance $A'B = \sqrt{898}$ without verifying if the required point of intersection lies within the permissible domain of the parameter.

By defining the $x$ -coordinate through a quadratic trigonometric expression, students are forced to first establish the global minimum of the coordinate ( $x \ge 12$ ).

Since the unconstrained geometric ideal ( $x \approx 10.38$ ) falls outside this interval, the true minimum occurs at the boundary constraint, penalizing pure geometric intuition that ignores algebraic bounds.

Quick Facts About This Question

Question Information

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