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Question #41647

Difficulty: 8/10 | 🟢 Free

The Question

A line $L$ lies in the plane $P: x + y - z = 0$ and is tangent to the surface $S: z = xy$ at a point in the first octant. Let $L$ intersect the planes $x = 0$ and $y = 0$ at points $A$ and $B$ respectively. Find the minimum value of $OA^2 + OB^2$ , where $O$ is the origin.

Options

A.
64
B.
32
C.
128
D.
16

Check Your Answer

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

Line lying in a plane|Condition of Tangency|Arithmetic Mean Inequality|Maximum/Minimum of Quadratic Expressions

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of 3D geometry with algebraic inequalities.

The lethal trap lies in the tangency condition: students often parameterize a generic line in the plane and attempt to minimize the distance globally, ignoring that tangency to the hyperbolic paraboloid $z=xy$ locks the line's intercepts into a strict constraint $(\sqrt{x_B}-1)(\sqrt{y_A}-1)=1$ .

The AM-GM inequality must be applied twice—first on the sum of fourth powers, and then on the reciprocal sum $k+1/k$ —which is only possible if the discriminant condition (tangency) is correctly translated into this parametric constraint.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Line lying in a plane
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free