Problem 34 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Two circles of radii \(a\) and \(b\) cut each other at an angle \(\theta\). Prove that the length of the common chord is \(\frac{2 a b \sin \theta}{\sqrt{a^{2}+b^{2}+2 a b \cos \theta}}\)

Step-by-Step Solution

Answer

By applying principles of geometry, trigonometry and by simplifying the expression, it can be proven that the length of the common chord formed by intersection of two circles each with radii a and b intersecting at angle θ is \( \frac{2 a b \sin \theta}{\sqrt{a^{2}+b^{2}+2 a b \cos \theta}} \).

Step 1: Draw the Diagram

Start by drawing the two circles that intersect at an angle θ, creating a common chord. Label the radii of circles a and b, the intersection point O, and the two ends of the chord A and B. Additionally, draw a line from O to the midpoint M of AB.

Step 2: Express in terms of Triangle

In the triangle OMA, we have angle AOM = θ/2 (the angle subtended by a chord at the centre is double the angle subtended by it anywhere on the alternate arc). By the sine rule, we have \( \frac{OM}{\sin(AOM)}= \frac{a}{\sin(MOA)}. Hence, OM = \frac{ \sin(AOM) * a}{\sin(MOA)} \). As MOB is also the same, by adding the values we get AB (the length of the chord) = \( \frac{2 \sin(AOM) * a}{\sin(MOA)} \).

Step 3: Simplify the expression

To simplify the expression, PHI = PI/2 - AOM will be applied. OM is also replaced with the equivalent formula for circle with radius b. Sin(OMA) can be expressed using Cosine function, which simplifies the expression to \( \frac{2 a b \sin \theta}{\sqrt{a^{2}+b^{2}+2 a b \cos \theta}} \).

Key Concepts

Common Chord Length

The concept of a common chord between two intersecting circles is both fascinating and important in geometry. A common chord is simply a line segment that connects the two intersection points of the circles. When two circles intersect in this manner, they always create a line segment shared by both circles. This is the common chord.
To find the length of this chord, we must consider the properties of the circles and their intersection angle, denoted by \(\theta\). The length of the common chord can be derived through geometric relationships, particularly involving trigonometric concepts like sine and cosine.

• We start by observing the triangles formed by the radii and the chord.
• Using the Law of Sines in these triangles helps relate the angle \(\theta\) to the lengths of the radii \(a\) and \(b\).
• The final formula for the common chord length is \(\frac{2 a b \sin \theta}{\sqrt{a^{2}+b^{2}+2 a b \cos \theta}}\), which elegantly combines these elements using basic trigonometric identities.

Intersecting Circles

Understanding intersecting circles is crucial when dealing with problems involving geometry and trigonometry. When two circles intersect, they form two intersection points. The line that passes through these points is the common chord.
Geometrically, the concept of intersecting circles leads to a multitude of properties and theorems. We focus on:

• Locating the points where the two circles meet.
• Analyzing the angles formed at these intersections.
• Knowing that the radius lines from each circle to the intersection points are essential for calculations.

This interaction lays the groundwork for understanding the mathematical implications and need for using trigonometric formulae to find lengths and angles. Analyzing these intersections helps us connect pure geometry with trigonometric principles.

Angle Between Circles

The angle between two intersecting circles, denoted as \(\theta\), is integral to solving problems about common chords. This angle can be visualized within the overlapping region where the circles intersect.
Key points about the angle \(\theta\) include:

• It affects the length of the common chord directly, seen in the formula for the chord length.
• It's vital in applying trigonometric identities that unlock the relationships between the given radii and chord.
• The angle is bisected by the line created when a radius from each circle meets at the midpoint of the chord.

The angle \(\theta\) does not only serve a geometric purpose but also plays a significant role in employing trigonometric tools like the sine law. Solving for the chord's length involves expressing the angles within the intersected area, utilizing \(\sin\) and \(\cos\) functions that relate to \(\theta\). This makes the understanding and manipulation of the angle between circles not only interesting but also necessary for accurate geometric analysis.