Problem 222 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

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

The common chord between the two intersecting circles has a length of \(\frac{2 a b \sin \theta}{\sqrt{a^{2}+b^{2}+2 a b \cos \theta}}\), derived using the law of cosines and trigonometric properties.

Step 1: Draw the Circles and Label Relevant Parts

First, draw the two intersecting circles with radii \(a\) and \(b\) and mark the point where they intersect, forming a chord. Label this chord as \(c\). Then form a triangle with the centers of the two circles and the middle point, \(M\), of the common chord. Let the angle between the lines joining the centres of the circles to \(M\) be \(\theta\). The length between the centres of the circles is \(d\) and it can be calculated as \(\sqrt{a^{2}+b^{2}+2ab\cos\theta}\).

Step 2: Calculate Half Length of Chord

Using right triangle properties, calculate the half length of the chord. This half-chord is the opposite side to the angle \(\theta\) in the triangle formed by the radii and the line joining the centres of the circles. Thus, the half-chord, \(OM\), is calculated as \(OM = b\sin\theta\).

Step 3: Double the Half-chord to Get the Chord

Now, double the length of half-chord to get the length of the whole chord \(c = 2*OM = 2b\sin\theta\).

Step 4: Substitute for b using earlier derived formula

Substitute for \(b\) using the formula we derived earlier, \(b = \sqrt{d^{2}-a^{2}}\), where \(d\) and \(a\) are the distances between the centres of the circles and the radii of the first circle. Plugging in the values we get \(c = 2\sqrt{d^{2}-a^{2}}\sin\theta\).

Step 5: Simplify the equation

Simplify the above equation to get the desired formula for the common chord length \(c =\frac{2 a b \sin \theta}{\sqrt{a^{2}+b^{2}+2 a b \cos \theta}}\). This completes the proof.

Key Concepts

Circle properties

In geometry, a circle is a simple yet profound shape. It consists of all points in a plane that are equidistant from a fixed point, known as the center. When examining the properties of a circle, we focus on several key elements:

• Radius (\( r \)): This is the distance from the center of the circle to any point on its boundary.
• Diameter: Twice the radius, the longest distance across the circle, passing through the center.
• Circumference: The total distance around the circle, calculated as \( 2\pi r \)
• Area: The space contained within the circle, given by \( \pi r^2 \)

These properties help us understand relationships within and between circles, like when two circles intersect, forming a common area. Understanding these basics lays the groundwork for delving into more complex problems involving circles, such as calculating the length of intersecting chords.

Chord length

A chord in a circle is a straight line segment whose endpoints lie on the circle. When two circles intersect, the line segment connecting their intersection points is called a common chord. Calculating the length of this chord involves a mix of circle properties and geometric relationships.For example, in our problem, the common chord length is derived using trigonometric relationships between the circle radii and the angle formed at the intersection. After constructing a triangle that incorporates these elements, the length of the common chord \( c \) is calculated using the formula:\[ c = \frac{2 a b \sin \theta}{\sqrt{a^{2}+b^{2}+2 a b \cos \theta}} \].This formula links the radius of each circle \( a \) and \( b \), the angle \( \theta \) between them, and the distance between their centers, providing a comprehensive view of the chord's length.

Trigonometry

Trigonometry, a branch of mathematics, focuses on the relationships between the sides and angles of triangles. It is crucial for solving problems involving circles, especially when angles and lengths need to be calculated.Key trigonometric ratios used in circle geometry include:

• Sine (\( \sin \)): Opposite side over hypotenuse in a right triangle.
• Cosine (\( \cos \)): Adjacent side over hypotenuse in a right triangle.
• Tangent (\( \tan \)): Opposite side over adjacent side in a right triangle.

In the case of intersecting circles, \( \sin \theta \) plays a significant role in determining the chord length. The formula uses trigonometric identities to express lengths and angles efficiently, showcasing how trigonometry bridges geometric concepts.

Intersection of circles

When two circles intersect, the points of intersection create a unique geometric figure. This configuration provides fascinating insights into geometry:

• Common Chord: The line segment connecting the two points of intersection.
• Spherical radius: Properties derived from each circle’s radius contribute to the characteristics of the intersecting figure.
• Central angle: Angle at which the chord subtends from the circle centers offers meaningful calculations.

In the exercise, intersecting circles are analyzed by considering an angle \( \theta \) formed at the intersection. This angle affects the length of the common chord. By understanding the interplay between these points and measurements, we gain deeper insights into circle-related geometry. The formula for the chord length incorporates changes in these elements, highlighting the importance of precise geometry in solving problems involving intersecting circles.