Problem 59 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If a circle of unit radius is divided into two parts by an arc of another circle subtending an angle \(60^{\circ}\) on the circumference of the first circle, then the radius of the arc is: (a) \(\sqrt{3}\) (b) \(\frac{1}{2}\) (c) 1 (d) \(\sqrt{2}\)

Step-by-Step Solution

Answer

The radius of the arc is 1 (option c).

Step 1: Understanding the Problem

We have a unit circle (radius = 1) and an arc of another circle that subtends an angle of 60 degrees on this unit circle. We are to find the radius of the second circle's arc.

Step 2: Visualize and Use Geometry

To find the radius of the second circle, visualize the setup. The arc from the second circle creates a sector angle of 60 degrees on the circumference of the first circle.

Step 3: Relate Arc Length and Angles

The length of an arc in a circle is given by \( \text{arc length} = r \theta \), where \( r \) is the radius of the circle and \( \theta \) is the central angle in radians. Here, \( \theta = \frac{60^{\circ}}{180^{\circ}} \times \pi = \frac{\pi}{3} \) radians.

Step 4: Equation for the Arc Length

Since the arc of the second circle subtends an angle on the first circle, its arc length equals the section on the first circle. So, for the second circle's arc, \( \text{arc length} = R \times \frac{\pi}{3} \).

Step 5: Set Up and Solve the Equation

Since it subtends an angle of \( \frac{\pi}{3} \) on the unit circle, the arc length is also equal to the arc length of the first circle: \( 1 \times \frac{\pi}{3} = R \times \frac{\pi}{3} \). Simplifying, \( R = 1 \times 1 = 1 \).

Step 6: Choose the Correct Option

From the calculations, the radius of the arc is 1, matching option (c).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry and mathematics as a whole. A unit circle is a circle with a radius of exactly one unit. Its center is located at the origin of a coordinate plane \((0,0)\). The unit circle simplifies many trigonometric calculations thanks to its consistent radius:

• Angles can be measured in radians, making it easy to convert between degrees and radians.
• The circle helps form a basis for defining sine, cosine, and tangent functions.

Since the radius is 1, any arc length in the unit circle is equal to the angle in radians.

Geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and the properties of space. In relation to circles, geometry provides us with tools to understand their various components, such as:

• Radius: The distance from the center to any point on the circle.
• Circumference: The total distance around the circle, calculated as \(2\pi r\).
• Arc: A segment or "piece" of the circle's circumference.
• Sector: A region bounded by two radii and the arc between them.

In the given exercise, geometry helps us visualize how the arc from one circle influences the unit circle by subtending an angle, which is key to solving the problem.

Arc Length

The arc length of a circle is crucial when dealing with sectors and angles. It represents the linear distance along the circle's circumference between two points. The formula to compute arc length is given by: \[\text{Arc Length} = r \cdot \theta\]where

• \(r\) is the radius of the circle.
• \(\theta\) is the central angle in radians.

For instance, in the problem, the arc length created by the second circle is determined using this formula, emphasizing the relationship between radius and subtended angle on a unit circle.

Central Angle

A central angle is formed by two radii that meet at the center of the circle, creating a specific measurable angle. In circle geometry, the central angle is directly linked to both the arc length and the sector area.

• Measured in degrees or radians.
• Such angles help determine the arc length of the circle and sector area.

In the exercise, we converted the central angle of \(60^{\circ}\) to radians (\(\frac{\pi}{3}\)) to find the radius. Understanding this conversion is essential for handling angular measures in trigonometry via the unit circle.