Before You Begin

Question #40809

Difficulty: 8/10 | 🟢 Free

The Question

A plane $P$ passes through the point $A(0, 0, 2)$ and is tangent to the ellipsoid $S: \frac{x^2}{3} + \frac{y^2}{2} + z^2 = 1$ . If the plane $P$ makes an angle $\theta$ with the plane $x - y = 0$ , then the maximum value of $\cos^2 \theta$ is

Options

A.
$\frac{11}{20}$
B.
$\frac{9}{20}$
C.
$\frac{3}{5}$
D.
$\frac{1}{2}$

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

Condition of Tangency|Angle between two planes|Direction Cosines constraint

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of 3D tangency conditions and vector normalization constraints.

The primary trap lies in treating the normal vector components of the plane as independent free variables.

Students often apply the algebraic condition of tangency ( $p^2 = a^2l^2 + b^2m^2 + c^2n^2$ ) but fail to couple it with the geometric direction cosines constraint ( $l^2+m^2+n^2=1$ ).

This omission leads to a breakdown when attempting to maximize the angle function, as the variables appear unbounded or the resulting derivative yields ghost roots.

By enforcing the DC constraint, the problem elegantly reduces to a constrained optimization solvable via the Cauchy-Schwarz inequality, bridging algebraic geometry with vector calculus.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Condition of Tangency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free