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Question #40913

Difficulty: 8/10 | 🟢 Free

The Question

Let $P$ be the unique point in the first quadrant that lies on both the conics $C_1: x^2 - y^2 - 8x + 4y + 12 = 0$ and $C_2: x^2 - 8y = 0$ . Let $A$ and $B$ be the orthogonal projections of $P$ onto the $x$ -axis and $y$ -axis, respectively. A ray of light emanating from $A$ strikes the parabola $y^2 = 16x$ at a point $Q$ and, after reflection, travels along a straight line that passes through $B$ . The length of the segment $AQ$ is:

Options

A.
$\frac{17}{4}$
B.
$\frac{15}{4}$
C.
$\frac{17}{2}$
D.
5

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Coordinate Axis Projections|Intersection of Conics|Reflective Property

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of degenerate conics and the geometric reflective properties of parabolas.

The primary trap lies in the reflection step: students who fail to recognize that the projection point A(4,0) is the exact focus of the parabola will attempt to use the general algebraic formula for the image of a point across a tangent line at Q(4t^2, 8t).

This brute-force method leads to a complex cubic equation (4t^3 - t^2 + 4t - 1 = 0) that is highly prone to calculation errors and consumes significant time.

Recognizing the focus geometrically forces the reflected ray to be parallel to the x-axis, bypassing the algebraic swamp entirely and rewarding geometric intuition.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Coordinate Axis Projections
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free