Problem 3 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A point is selected at random from the interior of a circle. The probability that the point is closer to the centre than the boundary of the circle is (A) \(\frac{3}{4}\) (B) \(\frac{1}{2}\) (C) \(\frac{1}{4}\) (D) none of these

Step-by-Step Solution

Answer

The probability is \(\frac{1}{4}\).

Step 1: Understanding the Problem

We need to find the probability that a randomly selected point from the interior of a circle is closer to the center than to the boundary.

Step 2: Recognizing Key Components

The interior of a circle consists of all points within the circle. We need to focus on determining the area that defines 'closer to the center' than to the boundary.

Step 3: Setting up the Problem

Let's consider a circle with center O and radius R. A point P within this circle would be closer to the center if its distance from O is less than \( \frac{R}{2} \).

Step 4: Determine Relevant Areas

The set of points closer to the center constitutes another circle with the same center O but with a radius of \( \frac{R}{2} \). The area of the original circle is \( \pi R^2 \) and the area of the smaller circle is \( \pi \left(\frac{R}{2}\right)^2 = \frac{\pi R^2}{4} \).

Step 5: Computing the Probability

The probability of a point being closer to the center is the ratio of the area of the smaller circle to that of the entire circle. This results in \( \frac{\frac{\pi R^2}{4}}{\pi R^2} = \frac{1}{4} \).

Key Concepts

Geometry: Understanding the Framework

Geometry is an essential branch of mathematics that deals with shapes, sizes, and the properties of space. Understanding geometry helps us investigate spatial relations and construct geometrical proofs. When dealing with problems involving probabilities in geometric settings, geometric principles guide us. For instance, identifying various sections of a shape, such as different areas within a circle, becomes crucial.
Understanding geometric properties aids in setting up problems for solutions, as in this exercise where we work with circles and their respective areas. Visualizing these concepts helps in comprehending spatial relationships and probability calculations.

Circle: Beyond the Basic Definition

A circle is more than just a round shape on paper; it's a set of points that are all equidistant from a center point called 'O'. One key feature of a circle is its radius, denoted as 'R', which is the distance from the center to any point on the boundary.
The problem illustrates a situation where we require knowledge of a circle's interior. This involves understanding that every point inside is closer to the center than most points on the boundary. To solve, recognize not just one circle but a smaller circle formed by points equidistant from the center within 'R/2'.

• The original circle has radius R
• The smaller circle, where points are closer to O, has radius R/2

This foundational understanding allows us to partition space into meaningful regions, essential for probabilistic calculations.

Probability Ratio: Devising Probable Outcomes

Probability measures how likely an event is to occur. When dealing with geometric shapes, like a circle, probabilities are often ratios of areas or lengths.
In this exercise, the probability that a point is closer to the circle's center than to its boundary relies on comparing two areas:

• The larger area is the entire circle (Area = πR²)
• The smaller area is the circle of radius R/2 (Area = π(R/2)² = πR²/4)

By forming the ratio of the smaller area to the larger, we calculate the probability:
\(\text{Probability} = \frac{\frac{\pi R^2}{4}}{\pi R^2} = \frac{1}{4} \)
Ratios render probabilities concrete by comparing meaningful sections of geometric figures.

Area Calculation: The Basis of the Solution

Calculating area is fundamental in geometry and is the cornerstone of determining probabilities in geometric contexts. The notion of area allows us to quantify how much space a shape occupies. Here, area makes a difference in probability problems involving a circle.To solve this problem:

• Calculate the area of the whole circle using the formula: \(\pi R^2\)
• Calculate the area of the smaller circle where points are closer to the center using: \(\pi \left( \frac{R}{2} \right)^2 = \frac{\pi R^2}{4}\)

These area calculations help us determine probabilities by providing a basis for comparing different parts of the circle. Through understanding areas, especially in a sphere or circle setting, abstract concepts of probability become perfectly tangible and clear.