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

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

(C) The locus is \(x^2 + y^2 = \frac{27}{4}\).

Step 1: Understanding the problem

We are asked to find the equation of the locus of the midpoints of chords in a circle with center \((0,0)\) and radius 3. These chords subtend an angle of \(\frac{2\pi}{3}\) at the center of the circle.

Step 2: Understanding the geometry of the circle

A chord of a circle will subtend an angle \(\theta\) at the center when its endpoints are separated by this angle. Here, \(\theta = \frac{2\pi}{3}\).

Step 3: Determine the length of the chord

The length of a chord that subtends an angle \(\theta\) at the center is given by the formula \(l = 2r \sin\left(\frac{\theta}{2}\right)\). With \(r = 3\) and \(\theta = \frac{2\pi}{3}\), we find \(l = 2 \times 3 \times \sin\left(\frac{\pi}{3}\right) = 3\sqrt{3}\).

Step 4: Finding the locus of midpoints

The midpoint of each chord will lie on a circle with smaller radius \(r'\), where \(r'\) is half of the chord's length. Therefore, \(r' = \frac{3\sqrt{3}}{2}\).

Step 5: Write equation of locus

Since the locus of the midpoints forms a circle centered at the origin with radius \(\frac{3\sqrt{3}}{2}\), the equation is \(x^2 + y^2 = \left(\frac{3\sqrt{3}}{2}\right)^2 = \frac{27}{4}\).

Key Concepts

Circle Geometry

Circle geometry involves studying various properties and relationships within a circle. A circle is a set of points equidistant from a central point, known as the center. The distance from the center of the circle to any point on its circumference is called the radius. Examples of the key elements in circle geometry include:

• The center, which is a fixed point from which all points on the circle are equidistant.
• The radius, a line segment joining the center and any point on the circle.
• The circumference, which is the distance around the circle.
• Chords, which are line segments with both endpoints on the circle.
• The diameter, the longest possible chord, which passes through the center.

In this exercise, we see the importance of understanding angles within circles, specifically the angle subtended by a chord at the center. This angle helps determine properties like the chord's length and refine other relationships between different circle components.

Chord Properties

Understanding the properties of chords is vital in circle geometry. A chord is any line segment with both endpoints on the circle. One crucial property of chords is that those subtending equal angles at the center have equal lengths. To explore chord properties further, consider:

• A chord subtends an angle at the center, which can be computed if the radius is known.
• The length of a chord, given by the formula: \[ l = 2r \sin\left(\frac{\theta}{2}\right) \]where \( r \) is the radius of the circle, and \( \theta \) is the subtended angle.
• If a chord subtends an angle of \( \frac{2\pi}{3} \) in a circle with a radius of 3, the length calculates to be \( 3\sqrt{3} \).
• The midpoint of a chord is always equidistant from the circle's center if the chords are of equal length.

These properties allow us to understand how chords interact within a circle and how their midpoints can form a new shape, such as another circle.

Locus Equation

A locus is the set of points satisfying a particular condition or rule. In the case of the locus of midpoints of chords, these points form a new circle. The locus equation describes this new shape. To find the locus equation of midpoints for chords:

• Determine the chord's midpoint; it lies equidistant from either endpoint.
• In the given exercise, a chord subtending an angle at the center helps determine the potential radius of this locus.
• The midpoint's path, or locus, forms a circle with a radius equal to half the length of the chord, here \( \frac{3\sqrt{3}}{2} \).
• This results in a new circle's equation, written as:
  \[ x^2 + y^2 = \left(\frac{3\cdot\sqrt{3}}{2}\right)^2 = \frac{27}{4} \]

This equation encapsulates the idea that all midpoints of the chords form another concentric circle, enriching our understanding of geometric relationships.