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Question #41762

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the line segment defined by the equation $x + y = 10$ for $5 \le x \le 8$ . Given two points $A(4,4)$ and $B(2,6)$ , find the minimum value of $PA + PB$ as the point $P$ varies over the segment $S$ .

Options

A.
4
B.
$\sqrt{2} + \sqrt{10}$
C.
$2\sqrt{5} + 2\sqrt{13}$
D.
$2\sqrt{10}$

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

Boundary Value Theorem|Condition of Collinearity|Reflection Principle

Why This Question Is Tricky

Common Mistakes

This question tests the Reflection Principle (Hero's problem) for minimizing distance sums, combined with the Boundary Value Theorem.

The trap is the collinearity condition: students will mechanically reflect point A, find the distance $A'B = 4$ , and select it as the minimum.

However, they fail to verify if the optimal collinear point $P(4,6)$ satisfies the constraints of the feasible segment ( $5 \le x \le 8$ ).

Because the unconstrained minimum lies outside the segment, the convexity of the distance function dictates that the true minimum occurs at the boundary point of the segment closest to the collinear point.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Boundary Value Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free