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Question #41784

Difficulty: 8/10 | 🟢 Free

The Question

Let the feasible region $R$ be defined by the inequalities $x^2 + 3y^2 \le 12$ , $x \ge 0$ , and $y \ge 0$ . A linear objective function $Z = ax + by$ (where $a, b > 0$ ) is to be maximized over $R$ . It is given that the maximum value of $Z$ is $12$ , and the line $ax + by = 12$ belongs to the family of lines $L_\lambda: (2+\lambda)x + (1-\lambda)y = 6$ . If the optimal solution occurs at the point $(x_0, y_0)$ , find the value of $a^2 + b^2 + x_0^2 + y_0^2$ .

Options

A.
28
B.
32
C.
24
D.
36

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

Corner Point Method Precision|Condition of Tangency|Family of Lines

Why This Question Is Tricky

Common Mistakes

This question tests a lethal trap in Linear Programming and Conic Sections.

Students habitually apply the Corner Point Method by checking only the algebraic vertices of the feasible region (e.g., $(2\sqrt{3}, 0)$ and $(0, 2)$ ).

If they assume the maximum occurs at a standard vertex, they will derive incorrect values for $a$ and $b$ that either violate the $a,b>0$ constraint or fail to bound the region correctly.

The problem forces a pivot to calculus and coordinate geometry: recognizing that for a non-linear boundary (ellipse), the optimal value occurs at a point of tangency where the objective function line touches the curve, rendering the standard vertex-checking algorithm incomplete.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Corner Point Method Precision
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free