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Question #40801

Difficulty: 8/10 | 🟢 Free

The Question

Let $E$ be the ellipse $\frac{x^2}{36} + \frac{y^2}{12} = 1$ in the $xy$ -plane, and let $C$ be its auxiliary circle. Let $L$ be the chord of $C$ joining the points having eccentric angles $0$ and $\frac{\pi}{2}$ . If $P$ is the pole of the line $L$ with respect to the ellipse $E$ , and $P'$ is the image of $P$ in the plane $\Pi: x + y + z = 14$ , find the square of the distance from $P'$ to the center of $C$ .

Options

A.
152
B.
208
C.
84
D.
48

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Ellipse Auxiliary Circle|Polar and Pole Duality|Image of a point in a plane

Why This Question Is Tricky

Common Mistakes

This question tests the duality of poles and polars combined with a 3D spatial transformation.

The primary trap lies in the dimensionality shift: students often find the 2D pole $P(6,2)$ correctly but fail to treat it as a 3D coordinate $P(6,2,0)$ when interacting with the plane $\Pi$ .

A common mistake is to project the plane onto 2D by setting $z=0$ , effectively finding the image of $(6,2)$ in the line $x+y=14$ , which yields the incorrect point $(12,8)$ and a squared distance of 208.

This forces the student to maintain rigorous spatial awareness when transitioning from planar conic sections to 3D analytical geometry.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Ellipse Auxiliary Circle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free