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Question #40576

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \{-5, -4, -3, -2, -1, 1, 2, 3, 4, 5\}$ . Find the number of ordered triplets $(a,b,c)$ such that $a \in A$ , $b \in A$ , $c \in \mathbb{Z}$ , the line $ax + by + c = 0$ is tangent to the ellipse $9x^2 + 4y^2 = 36$ , and the distance between this line and its parallel tangent is $2\sqrt{5}$ .

Options

A.
4
B.
8
C.
16
D.
32

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

Selection vs Arrangement|Condition of Tangency|Distance Between Parallel Lines

Why This Question Is Tricky

Common Mistakes

This question tests the distinction between algebraic arrangements (ordered triplets) and geometric existence (distinct lines), while utilizing the tangency condition of an ellipse.

The trap is triggered by the distance constraint: students often solve for the relationship between the coefficients ( $a = \pm 2b$ ) and count the 8 valid pairs of $(a,b)$ .

However, if they perform a pure 'arrangement' of these coefficients without verifying the geometric reality of the parallel tangent possibilities ( $c = \pm 5b$ ), they miss the 'hidden' pairs created by the dual-tangent symmetry of the conic section, leading to a deficit in the final count (answering 8 instead of 16).

Conversely, counting only distinct geometric lines yields 4, punishing those who ignore the algebraic permutation aspect.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Selection vs Arrangement
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free