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Question #40964

Difficulty: 8/10 | 🟢 Free

The Question

A composite reflective surface is formed by the parabola $y^2 = 4(x+4)$ for $x < 0$ and the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ for $x \ge 0$ . A light ray is emitted from the point $S(-3, 0)$ and strikes the parabolic surface at a point $P$ in the first quadrant (where $y > 0$ ). The ray reflects off the parabola, travels parallel to the x-axis, and then strikes the elliptical surface at a point $Q$ . The ray reflects off the ellipse and intersects the x-axis at a point $R$ . Find the x-coordinate of $R$ .

Options

A.
$\frac{127\sqrt{3}}{60}$
B.
$3$
C.
$\frac{150\sqrt{3}}{23}$
D.
$\frac{127}{60}$

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

Physical Interpretation|Reflective Property|Intersection of Conics

Why This Question Is Tricky

Common Mistakes

This question tests the physical interpretation of reflective properties across a composite domain of intersecting conics.

The primary trap lies in the false assumption of symmetry: students often correctly identify that the parabola and ellipse share the focus $S(-3,0)$ , and knowing that the ellipse's other focus is at $(3,0)$ , they incorrectly assume that a ray traveling parallel to the major axis will reflect through the focus $(3,0)$ (Option B).

However, the reflective property of an ellipse only directs rays to the opposite focus if they originate from the first focus.

A ray parallel to the major axis does not pass through the focus of an ellipse, requiring students to manually re-derive the normal vector using implicit differentiation at the specific intersection coordinate $Q$ rather than relying on generic focal properties.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Physical Interpretation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free