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Question #41550

Difficulty: 8/10 | 🟢 Free

The Question

A curve $C_1$ passing through the origin has the property that any ray of light moving parallel to the $x$ -axis reflects off $C_1$ and passes through the point $S(1,0)$ . Let $C_2$ be an ellipse whose foci are the origin and $S$ , and whose major axis has length $2$ . If $A$ is the area of the region bounded by $C_1$ and $C_2$ containing the point $\left(\frac{1}{2}, 0\right)$ , then $A$ equals:

Options

A.
$\frac{5\sqrt{6}}{27} + \frac{\pi\sqrt{3}}{4} + \frac{\sqrt{3}}{2}\sin^{-1}\left(\frac{1}{3}\right)$
B.
$\frac{5\sqrt{6}}{54} + \frac{\pi\sqrt{3}}{8} + \frac{\sqrt{3}}{4}\sin^{-1}\left(\frac{1}{3}\right)$
C.
$-\frac{\sqrt{6}}{27} + \frac{\pi\sqrt{3}}{4} - \frac{\sqrt{3}}{2}\sin^{-1}\left(\frac{1}{3}\right)$
D.
$\frac{5\sqrt{6}}{27} + \frac{\pi\sqrt{3}}{4} - \frac{\sqrt{3}}{2}\sin^{-1}\left(\frac{1}{3}\right)$

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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

Symmetry Utilization|Reflective Property|Intersection of Conics

Why This Question Is Tricky

Common Mistakes

This problem tests the translation of physical reflective properties into geometric loci to identify the parabola.

The primary trap lies in the intersection of the two conics, which yields a quadratic in $x$ with an extraneous negative root $x = -9/2$ .

Students often fail to discard this root, leading to incorrect integration limits.

Furthermore, the integration requires splitting the domain at the correct intersection point $x = 1/6$ and carefully managing the sign of the lower limit in the trigonometric substitution.

Forgetting the symmetry multiplier of 2 is another classic pitfall designed to catch rushed calculations.

Quick Facts About This Question

Question Information

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