Problem 122 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(C\) be the circle with centre \((0,0)\) and radius 3 units. The equation of the locus of the mid points of the chords of the circle \(C\) that subtend an angle of \(\frac{2 \pi}{3}\) at its centre is (A) \(x^{2}+y^{2}=\frac{3}{2}\) (B) \(x^{2}+y^{2}=1\) (C) \(x^{2}+y^{2}=\frac{27}{4}\) (D) \(x^{2}+y^{2}=\frac{9}{4}\)

Step-by-Step Solution

Answer

(D) \(x^{2}+y^{2}=\frac{9}{4}\)

Step 1: Understand the Problem

We need to find the equation of the locus of midpoints of chords that subtend a given angle at the center of a circle. The center of the circle is at the origin (0,0) with a radius of 3 units, and the angle subtended by the chord at the center is \(\frac{2\pi}{3}\).

Step 2: Determine the Perpendicular Distance to Chord

The perpendicular distance from the center to a chord that subtends an angle \( \theta \) at the center is given by \( R \cos\left(\frac{\theta}{2}\right) \). Here, \( R = 3 \) and \( \theta = \frac{2\pi}{3} \), so the perpendicular distance is \( 3 \cos\left(\frac{\pi}{3}\right) = 3 \cdot \frac{1}{2} = \frac{3}{2} \).

Step 3: Find the Locus Equation

For the circle \( x^2 + y^2 = R^2 \), the locus of the midpoints of these chords is a circle centered at the origin with a radius equal to the perpendicular distance from the center of the main circle to the chord. Since the perpendicular distance is \( \frac{3}{2} \), the locus of the midpoints of the chords is \( x^2 + y^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \).

Step 4: Verify Suitable Options

Compare \( x^2 + y^2 = \frac{9}{4} \) with the given options. The correct option is \( D \).

Key Concepts

Understanding Geometry

Geometry is a fascinating branch of mathematics dealing with the properties and relations of points, lines, surfaces, and solids. It helps us explore the spatial configurations of objects. In this exercise, we are dealing with a circle - which is a fundamental geometric shape. Understanding circles is crucial in geometry as they appear frequently in different contexts.A circle is defined by its center and radius:

• The **center** is a point inside the circle. In this problem, it is the point \(0, 0\), which means the circle is centered at the origin of a coordinate plane.
• The **radius** is the distance from the center to any point on the circle. Here, the radius is 3 units.

For a chord, which is a line segment within the circle connecting two points on it, understanding its properties is essential. The midpoints of such chords can form interesting patterns, called loci, which are other shapes or points that we can explore using geometry.

Circle Equations Basics

Circle equations are tools that represent the relationship between the circle's radius and points on its boundary. The general equation for a circle centered at origin \(0, 0\) is \(x^2 + y^2 = R^2\), where \(R\) is the radius.In this task, the circle's equation is given by \(x^2 + y^2 = 9\) because the radius is 3 units (since \(R = 3\), thus \(R^2 = 9\)).Moreover, knowing the concept of loci within circle equations can lead to new shapes. A locus is a collection of points that shares a particular property. Here, the locus of midpoints of chords reflecting certain angular properties forms another interesting shape around the origin.By calculating and understanding the perpendicular distance from the circle center to the chord, we can define another circle (locus) equation. This new circle's equation is determined by the intersection and distances pertaining to the original circle.

Angle Subtended by a Chord

Angles subtended by a chord are a critical aspect of circle geometry. When a chord subtends an angle at the center of the circle, it creates an intersection of intriguing geometric principles.In this problem, we're considering chords that subtend an angle of \(\frac{2\pi}{3}\) radians. This angle at the circle's center is key to determining the locus. By leveraging the angle, we can find out how far the locus of midpoints will likely form from the center by calculating the perpendicular distance.

• The formula \( R \cos(\frac{\theta}{2})\) helps us find this distance, where \(R\) is the circle radius and \(\theta\) is the subtended angle.
• In this exercise, the perpendicular distance from the circle's center to the chord is \(\frac{3}{2}\), derived from \(3 \cdot \cos(\frac{\pi}{3}) = \frac{3}{2}\).

Thus, understanding how a chord's subtended angle interacts with these properties helps us perceive the overarching geometric patterns within and around circles.