Problem 77 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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 center 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 locus of the midpoints of chords of a circle with radius 3 that subtend an angle of \(\frac{2\pi}{3}\) at the center of the circle \((0,0)\).

Step 2: Identify the Properties of the Chords

The angle subtended by chord at the circle's center helps identify properties of the chord. When a chord subtends an angle of \(\theta\), its length is given by \(l = 2r \sin\left(\frac{\theta}{2}\right)\). Here, \(r = 3\) and \(\theta = \frac{2\pi}{3}\).

Step 3: Calculate the Length of the Chord

Substitute the known values into the chord length formula:\[ l = 2 \times 3 \times \sin\left(\frac{\frac{2\pi}{3}}{2}\right) \]\[ l = 6 \times \sin\left(\frac{\pi}{3}\right) \]\[ l = 6 \times \frac{\sqrt{3}}{2} \]\[ l = 3\sqrt{3} \]

Step 4: Find the Locus of Midpoints

A midpoint of a chord is half of the chord's distance away from the center on the perpendicular bisector. The locus of such midpoints forms a circle, centered at the origin \((0,0)\), with radius \(r \cos\left(\frac{\theta}{2}\right)\).

Step 5: Calculate the Radius of Locus Circle

Using our chord property:\[ r' = 3 \cos\left(\frac{\pi}{3}\right) \]\[ r' = 3 \times \frac{1}{2} \]\[ r' = \frac{3}{2} \]

Step 6: Write Equation of the Locus

Since the center is (0,0) and radius calculated is \(\frac{3}{2}\), the equation for the locus is:\[ x^{2} + y^{2} = \left(\frac{3}{2}\right)^{2} \]\[ x^{2} + y^{2} = \frac{9}{4} \]

Key Concepts

Circle Geometry

Circle geometry is a fascinating branch of mathematics that deals with the properties and relationships of circles. A circle is defined by its center point and a constant radius, which is the distance from the center to any point on the circle. In our exercise, the circle has its center at (0, 0) and a radius of 3 units. Circles are important in many areas of mathematics and are defined by the equation: \[ x^{2} + y^{2} = r^{2} \] where \(r\) represents the radius. This equation tells us that any point \((x, y)\) that satisfies it will lie on the circle. Knowing how to find points on or related to the circle, like the locus of midpoints, is an essential skill while working with circle geometry."},{

Angle Subtended by Chord

When studying circles, the angle subtended by a chord at the center is an exciting concept. A chord is a line segment with both endpoints on the circle. In our problem, a chord subtends an angle of \(\frac{2\pi}{3}\) at the center, meaning the endpoints of the chord lie such that the angle between lines from the center to the endpoints is \(\frac{2\pi}{3}\).This angle helps us determine specific properties of the chord, like its length or influence on other facets of the circle. The chord's length, given the subtended angle \(\theta\) and radius \(r\), is calculated by:\[ l = 2r \sin\left(\frac{\theta}{2}\right) \]In our case, substituting \(r = 3\) and \(\theta = \frac{2\pi}{3}\) into this formula gives us a chord length of \(3\sqrt{3}\). Understanding this subtended angle is key to finding how the chord interacts with the circle and its features.

Locus Equation

The concept of a locus is crucial in understanding the behavior of points within geometric spaces, like circles. A locus is a collection or set of points satisfying a particular condition or a defined rule. In this problem, we are interested in the locus of midpoints of the chords that subtend a specific angle at the circle's center. These midpoints will form another circle within the original circle. This derived circle, called the locus, has a center at the origin since it maintains symmetry relative to the larger circle. The radius of this locus can be calculated using the cosine of half the subtended angle and the original circle's radius:\[ r' = r \cos\left(\frac{\theta}{2}\right) \]With our values, we find \(r' = \frac{3}{2}\). The locus equation is then:\[ x^{2} + y^{2} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4} \]This equation describes the set of all possible midpoints of the chords under the given conditions, highlighting the elegance and interconnectedness of circle geometry.

Radius of Circle

The radius of a circle is a fundamental concept in geometry, representing the distance from the center to any point on the circle. This fixed measure is pivotal in equations involving circles, determining the size and influence of the circle in geometric calculations.In the given exercise, the radius is a vital component when calculating chord properties and the locus of midpoint. The original circle has a radius of 3 units, a simple yet important value. Using the radius, we determine both chord lengths and the radius of the locus.When constructing the locus circle, the radius of this secondary circle was identified using a segment of our previous calculation involving \(\cos(\theta/2)\). This method showed how beatifully radius-related calculations extend into more complex tasks, such as computing loci.Being adept at utilizing the radius ensures you understand not just basic circle properties, but also appreciate its applications in more comprehensive problems within circle geometry.