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Question #40837

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the conic given by $x^2 - y^2 - 2x + 4y - 4 = 0$ . A variable line $L: lx + my = 1$ intersects $S$ at two distinct points $P$ and $Q$ . The position vectors $\vec{p}$ and $\vec{q}$ of $P$ and $Q$ with respect to the origin $O$ are linearly independent, and the scalar projection of $\vec{p}$ along $\vec{q}$ is zero.

As $L$ varies, it envelopes a curve $E$ . The altitudes of the triangle formed by any three such lines $L$ are concurrent at a point $H$ . As the three lines vary, the locus of $H$ is a straight line $D$ . Which of the following points lies on $D$ ?

Options

A.
$(-4, 0)$
B.
$(0, 4)$
C.
$(2, -1)$
D.
$(4, 2)$

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

Intersection of Conics|Concurrent Lines Condition|Linear Independence vs Dependence|Dot Product Projection Sign

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of conics via homogenization and the geometric interpretation of linear dependence and dot products.

The primary trap lies in the dual space transformation: students often mechanically apply the perpendicularity condition to get $l^2+m^2+l/2-m=0$ , but fail to recognize this as the inverse pedal transformation.

Assuming the envelope is a complex higher-degree curve instead of recognizing that a linear pedal curve implies a parabola is a common pitfall.

Furthermore, the concurrency of altitudes (the orthocenter) of the tangent triangle tests the advanced property that it lies on the directrix.

A sign error in the dot product projection or directrix distance scaling will lead to incorrect parallel lines, trapping students in the incorrect options.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Intersection of Conics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free