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Question #41798

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{(h, k) \in \mathbb{Z} \times \mathbb{Z} : h^2 + k^2 \le 34\}$ . A point $(h, k)$ is chosen at random from $S$ . Given that the equation $hx^2 - hy^2 + 2kxy + 2x - 2y + 1 = 0$ represents a conic section, what is the probability that it represents a non-degenerate rectangular hyperbola?

Options

A.
$\frac{35}{36}$
B.
$\frac{105}{109}$
C.
$\frac{104}{109}$
D.
$\frac{26}{27}$

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

Sample Space Partitioning|Rectangular Hyperbola|Locus of a Moving Point

Why This Question Is Tricky

Common Mistakes

This question tests the precise definition of a conic section and the conditions for a rectangular hyperbola.

The primary trap is failing to adjust the sample space denominator.

Students often mechanically calculate the total number of points (109) and use it as the denominator, forgetting that at $(0,0)$ the equation degenerates to a straight line (degree 1), which is not a conic section.

Furthermore, they must identify that $\Delta = 0$ yields a pair of perpendicular lines (a degenerate rectangular hyperbola), which must be excluded from the numerator to satisfy the 'non-degenerate' condition.

Failing to exclude $(0,0)$ from the denominator leads to the trap answer 105/109, while miscounting the degenerate cases leads to 104/109 or 26/27.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Sample Space Partitioning
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free