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Question #41743

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the quadric surface given by $x^2 + y^2 + z^2 - 2xy - 2yz - 2zx = 0$ . A line $L$ , given in non-symmetric form by the intersection of the planes $px + qy = 102$ and $z = 270$ , is tangent to the surface $S$ . If $p$ and $q$ are positive real numbers, what is the minimum value of $p+q$ ?

Options

A.
$\frac{68}{45}$
B.
$\frac{34}{45}$
C.
$\frac{45}{17}$
D.
$\frac{17}{45}$

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

Symmetric vs Non-symmetric form|Condition of Tangency|Concavity Analysis|Harmonic Mean Constraints

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of a quadric surface with a plane, reducing the 3D geometry to a 2D conic (parabola) tangency.

The tangency condition elegantly collapses into a harmonic mean constraint ( $1/p + 1/q = 45/17$ ) via the adjugate matrix method.

A common trap is misinterpreting the 3D concavity and assuming the line must be a ruling (which doesn't exist for this section), or failing to recognize that the tangency point is exactly the vertex of the parabolic section, which acts as a local extremum in the restricted plane.

Students who blindly substitute without analyzing the discriminant matrix often get lost in massive algebraic expansions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Symmetric vs Non-symmetric form
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free