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Question #40814

Difficulty: 8/10 | 🟢 Free

The Question

Let $E$ be the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and $T$ be the tangent to $E$ at the point $A(0, 3)$ . Let $P$ be a finite point in the $xy$ -plane such that its chord of contact with respect to $E$ is parallel to $T$ . If this chord of contact intersects $E$ in two distinct points, and $d$ is the distance between $T$ and the chord of contact, then the set of all possible values of $d$ is:

Options

A.
$(0, 6)$
B.
$(0, 3) \cup (3, 6)$
C.
$(0, 6]$
D.
$[0, 3) \cup (3, 6]$

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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

Chord of Contact Logic|Distance Between Parallel Lines|Closed vs Open Intervals

Why This Question Is Tricky

Common Mistakes

This question tests the geometric understanding of the chord of contact and its behavior as the external point moves.

A common trap is assuming the chord of contact can sweep the entire area between the boundary tangents, yielding a continuous interval $(0, 6)$ for the distance.

However, for a central conic, the chord of contact passes through the center only when the external point is at infinity (the center is the pole of the line at infinity).

Since the point is explicitly defined as finite, the chord can never be the major/minor axis.

This excludes the distance $d=3$ , splitting the interval and testing the student's awareness of open intervals and points at infinity.

Quick Facts About This Question

Question Information

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