Problem 58 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A square is inscribed in a circle. If \(p_{1}\) is the probability that a randomly chosen point of the circle lies within the square and \(p_{2}\) is the probability that the point lies outside the square then(A) \(p_{1}=p_{2}\) (B) \(p_{1}>p_{2}\) and \(p_{1}^{2}-p_{2}^{2}<\frac{1}{3}\) (C) \(p_{1}

Step-by-Step Solution

Answer

(C) \(p_1 < p_2\)

Step 1: Understand the Problem

We have a square inscribed in a circle. We're asked to find the relationship between two probabilities: \(p_1\), the probability of a point being inside the square, and \(p_2\), the probability of it being outside the square.

Step 2: Define Variables and Relationships

Let the side of the square be \(s\). The diameter of the circle, which is also the diagonal of the square, is \(s\sqrt{2}\). Hence, the radius of the circle \(r\) is \(\frac{s\sqrt{2}}{2}\).

Step 3: Calculate Area of the Square and Circle

The area of the square is \(s^2\) and the area of the circle is \(\pi\left(\frac{s\sqrt{2}}{2}\right)^2 = \frac{\pi s^2}{2}\).

Step 4: Calculate Probability \(p_1\)

The probability \(p_1\) that a point lies within the square is the ratio of the area of the square to the area of the circle: \(p_1 = \frac{s^2}{\frac{\pi s^2}{2}} = \frac{2}{\pi}\).

Step 5: Calculate Probability \(p_2\)

The probability \(p_2\) that a point lies outside the square is the complement of \(p_1\): \(p_2 = 1 - p_1 = 1 - \frac{2}{\pi}\).

Step 6: Compare \(p_1\) and \(p_2\)

Since \(\frac{2}{\pi} \approx 0.636 < 0.5\), it follows that \(p_1 < p_2\). Therefore, (C) \(p_1 < p_2\) is the correct answer.

Key Concepts

Inscribed Shapes

When dealing with inscribed shapes, it's common to encounter a scenario where a square is snugly fit inside a circle, touching the circle at exactly four points on its circumference. This situation occurs because the square's diagonal aligns perfectly with the diameter of the circle. This alignment forms the basis for many geometric probability problems. Inscribed shapes are a staple in geometry, where one shape is contained entirely within another. In our specific problem, the square is inscribed within a circle. This means:

• The vertices (corners) of the square touch the circle.
• The diagonal of the square is the same length as the diameter of the circle.

Understanding inscribed shapes allows us to calculate various geometric properties, such as areas and probabilities, based on the relationships between the involved shapes.

Circle and Square Relationship

The relationship between a circle and an inscribed square is fundamental to solving problems related to geometry and probability. For an inscribed square inside a circle, the diameter of the circle is equal to the diagonal of the square. This is a key fact used to establish relationships between different dimensions.Imagine a square with side length \(s\), inscribed in a circle. The diagonal of this square is calculated using the Pythagorean theorem as \(s\sqrt{2}\). Since the diagonal of the square is also the diameter of the circle, we can deduce that the circle's radius \(r\) is given by \(r = \frac{s\sqrt{2}}{2}\).This relationship is crucial for determining geometric probabilities, as it allows for calculations of areas and other important measures. Knowing how these two shapes align within each other simplifies solving related problems significantly.

Area Calculation

Calculating areas of geometric shapes is essential when determining probabilities related to these shapes. For our problem, we need to find the areas of both the square and the circle in order to compute the probabilities.First, the area of the square is straightforward:

• Area of the square is \(s^2\), where \(s\) is the side length.

For the circle, since the radius \(r\) has been determined as \(\frac{s\sqrt{2}}{2}\), the area of the circle is computed as:

• Area of the circle = \(\pi \left(\frac{s\sqrt{2}}{2}\right)^2 = \frac{\pi s^2}{2}\).

These areas give rise to the probabilities. The probability \(p_1\) of a point being inside the square is the square's area divided by the circle's area, thus \(p_1 = \frac{2}{\pi}\). Meanwhile, the probability \(p_2\) of a point being outside the square complements \(p_1\), determined by \(p_2 = 1 - \frac{2}{\pi}\). These calculations are fundamental to evaluating the likelihoods of a point lying within certain geometric regions.