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Question #41489

Difficulty: 8/10 | 🟢 Free

The Question

Let $P_1$ be the plane passing through the point $(1, 1, 1)$ and the line of intersection of the planes $x + y + z = 1$ and $x - y + z = 3$ . Let $C$ be the solid cylinder $x^2 + y^2 \le 4$ . The plane $P_1$ intersects the cylinder $C$ in an elliptical region $R$ . If $A$ is the area of the orthogonal projection of the region $R$ onto the plane $x + y + \sqrt{2}z = 0$ , then find $A$ .

Options

A.
$2\pi(\sqrt{2} + 1)$
B.
$\pi(\sqrt{2} + 2)$
C.
$\pi(\sqrt{2} + 1)$
D.
$4\pi(\sqrt{2} + 1)$

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

Intersection Point Accuracy|Equation of plane through intersection|Projection vs. Coordinate

Why This Question Is Tricky

Common Mistakes

This question tests the 'Intersection Point Accuracy' and 'Projection vs.

Coordinate' traps in 3D geometry.

Students often incorrectly assume the projection of the intersection curve can be found by directly projecting the cylinder's base area onto the target plane, bypassing the intermediate plane $P_1$ (leading to Option B).

Another common error is multiplying by the cosine factor instead of dividing when finding the true area of the elliptical region $R$ (leading to Option C).

The problem forces the sequential application of the area projection formula $A' = A \cos \theta$ using correct normal vectors, rather than confusing the projection length with the actual coordinate distance in 3D space.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Intersection Point Accuracy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free