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Question #40815

Difficulty: 8/10 | 🟢 Free

The Question

A fair coin is tossed. If it lands heads, a point $P(x_0, y_0)$ is chosen uniformly at random from the set $S_1 = \{(x,y) \in \mathbb{Z}^2 \mid \max(|x|,|y|) \le 2\}$ . If it lands tails, $P$ is chosen uniformly at random from the set $S_2 = \{(x,y) \in \mathbb{Z}^2 \mid \max(|x|,|y|) = 3\}$ . It is observed that a pair of distinct real tangents can be drawn from $P$ to the circle $x^2+y^2=4$ , and the slope of their chord of contact is an integer. What is the probability that the coin landed heads?

Options

A.
$\frac{96}{221}$
B.
$\frac{192}{317}$
C.
$\frac{4}{9}$
D.
$\frac{24}{49}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Chord of Contact Logic|Combinatorial Probability|Bayes' Theorem Reversal

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of coordinate geometry constraints with conditional probability and Bayes' Theorem.

The primary trap lies in the restricted sample space: students often compute the probability of an integer slope over all points in $S_1$ , forgetting that the existence of 'distinct real tangents' mandates $x_0^2 + y_0^2 > 4$ .

This filters out points like $(0, \pm 1)$ or $(\pm 2, 0)$ which might otherwise yield integer slopes (leading to the trap answer 192/317).

Furthermore, the two-stage selection process (coin toss followed by point selection from sets of different sizes) forces a rigorous application of Bayes' Theorem, penalizing those who incorrectly assume a uniform distribution over $S_1 \cup S_2$ (leading to the trap answer 4/9).

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