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Question #40767

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1, L_2$ , and $L_3$ be the lines given by $3x - 4y = 0$ , $4x + 3y - 25 = 0$ , and $y = 0$ respectively. Let $S$ be the set of all points $P$ on the curve $C: 4x^2 + 12y - 25x = 0$ such that the reflections of $P$ across $L_1, L_2$ , and $L_3$ yield exactly three distinct points which are collinear. Find the number of elements in $S$ .

Options

A.
0
B.
1
C.
2
D.
4

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

Reflection Principle|Condition of Collinearity|Extraneous Solution Rejection

Why This Question Is Tricky

Common Mistakes

This question tests the geometric Reflection Principle (Steiner's Line Theorem) combined with Extraneous Solution Rejection.

Students must recognize that collinearity of reflections across triangle sides requires the point to lie on the circumcircle.

The trap lies in the intersection of the given curve and the circumcircle, which yields four points.

Three of these points are the vertices of the triangle.

A common mistake is to accept all four points, forgetting that the reflection of a vertex across its adjacent sides maps to the same point, violating the 'three distinct points' condition.

Rejecting these extraneous geometric solutions is crucial for the correct answer.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Reflection Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free