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Question #41746

Difficulty: 8/10 | 🟢 Free

The Question

Let $E$ be the ellipse formed by the intersection of the ellipsoid $5x^2 + 6xy + 5y^2 + 8z^2 = 72$ and the plane $x + y + z = 0$ . Let $C$ be the auxiliary circle of $E$ lying in the same plane. For any point $P$ on $C$ , let $P'$ be the corresponding point on $E$ such that $P$ and $P'$ have the same eccentric angle with respect to the principal axes of $E$ . Let $L$ be the maximum length of the orthogonal projection of the line segment $PP'$ onto the $z$ -axis as $P$ varies on $C$ . Find the value of $L^2$ .

Options

A.
0
B.
3
C.
6
D.
9

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

Orthogonal projection of a line|Ellipse Auxiliary Circle|Coordinate Axis Projections|Triangular/Diagonal Matrix Shortcut

Why This Question Is Tricky

Common Mistakes

This question tests the geometric interpretation of 3D-rotated conic sections and their auxiliary circles.

The primary trap lies in the block-diagonal structure of the quadratic form matrix.

Students often incorrectly assume that the z-axis decouples from the xy-plane, leading them to project the ellipsoid directly onto the coordinate planes without accounting for the coupling introduced by the intersecting plane x+y+z=0.

Furthermore, the problem requires recognizing that the vector connecting corresponding points on an ellipse and its auxiliary circle is always parallel to the minor axis, forcing the computation of the exact 3D orientation of this axis rather than just its scalar length.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Orthogonal projection of a line
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free