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Question #73316

Difficulty: 8/10 | 🟢 Free

The Question

Let $\hat{u} = l\hat{i} + m\hat{j} + n\hat{k}$ be a unit vector such that $n=0$ . A point $P(al, bm)$ lies on the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ( $a > b > 0$ ) and $\vec{v}$ is the position vector of $P$ .

The points $P$ are constrained by the modulus inequality: $|\vec{v}| \le \sqrt{\frac{a^2+b^2}{2}}$

Let $Q(al, am)$ be the point on the auxiliary circle of $E$ corresponding to $P$ . The region $R$ is the union of the sectors of the auxiliary circle subtended at the origin by all valid points $Q$ .

If the area of $R$ is $18\pi$ and the eccentricity of $E$ is $\frac{\sqrt{5}}{3}$ , find the area of the ellipse $E$ .

Options

A.
$24\pi$
B.
$36\pi$
C.
$96\pi$
D.
$12\pi$

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

Direction Cosines constraint|Ellipse Auxiliary Circle|Modulus Inequality Logic

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the intersection of Coordinate Geometry and Vector Algebra through a Modulus Inequality Trap.

Students often misinterpret the magnitude constraint $|\vec{v}| \le k$ by attempting to find the area of the intersection of the ellipse and a circle, rather than mapping the constraint to the parametric eccentric angle $\theta$ .

The trap forces a restricted domain ( $\cos^2\theta \le 1/2$ ).

A common lethal mistake is calculating the sector area on the ellipse ( $\frac{1}{2}ab\Delta\theta$ ) instead of the requested auxiliary circle ( $\frac{1}{2}a^2\Delta\theta$ ), leading to an incorrect semi-major axis ( $a = 3\sqrt{6}$ ) and ultimately the wrong ellipse area of $36\pi$ (Option B).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Direction Cosines constraint
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free