Problem 24 - A Complete Resource Book in Mathematics for JEE Main

Question

If two points \(A(a, 0)\) and \(B(-a, 0)\) are stationary and if \(\angle A-\angle B=\theta\) in \(\triangle A B C\), the locus of \(C\) is (A) \(x^{2}+y^{2}+2 x y \tan \theta=a^{2}\) (B) \(x^{2}-y^{2}+2 x y \tan \theta=a^{2}\) (C) \(x^{2}+y^{2}+2 x y \cot \theta=a^{2}\) (D) \(x^{2}-y^{2}+2 x y \cot \theta=a^{2}\)

Step-by-Step Solution

Answer

The correct equation is (A) \(x^2 + y^2 + 2xy \tan \theta = a^2\).

Step 1: Understanding the Problem

We are asked to find the locus of point \(C\) in a triangle where the fixed points \(A(a, 0)\) and \(B(-a, 0)\) affect the angles \(\angle A\) and \(\angle B\) such that the difference is \(\theta\). The task is to determine which of the provided equations represents this locus.

Step 2: Use of Angle Property

Since \(\angle A - \angle B = \theta\), we can use the property of angular bisectors. The angular bisector divides angles equally, and the difference between the angles at points \(A\) and \(B\) implies a connection of point \(C\) having a special geometric relationship determined by the angles.

Step 3: Applying Trigonometric Functions

The cotangent and tangent of the angle \(\theta\) are often used in the context of these types of problems. We should consider expressions involving \(x\), \(y\), and the tangent or cotangent of \(\theta\), given the properties of angles in triangles.

Step 4: Setting Up the Locus Equation

Considering the angle property \(\angle A - \angle B = \theta\), the locus relates to the angle bisector properties and can be expressed using tangent and cotangent identities. Generally, for a rotated angle, the equation \(x^2 + y^2 + 2xy \tan \theta = a^2\) or a similar variant will emerge from the geometric properties.

Step 5: Determining the Correct Equation

Using the previous investigation into the geometry and angles, the equation \(x^2 + y^2 + 2xy \tan \theta = a^2\) fits with the properties of angle bisectors for \(\angle A - \angle B = \theta\). Thus, this expresses the relationship of the locus for point \(C\).

Key Concepts

Trigonometric Functions

In mathematics, trigonometric functions relate the angles of triangles to the lengths of their sides. These functions include sine, cosine, tangent, cotangent, secant, and cosecant, but in the context of this problem, we focus on tangent and cotangent. These often appear when dealing with geometric relationships and angles within triangles.

Tangent (\( \tan \theta \) is the ratio of the opposite side to the adjacent side in a right triangle. While cotangent (\( \cot \theta \) is the reciprocal, being equal to the adjacent side over the opposite side. These functions help express relationships between angles and sides, making them particularly useful in solving problems involving triangles and angular relationships.

In the given problem, trigonometric functions like tangent and cotangent are utilized to express the locus equation of point \( C \). This involves identifying how angles in triangle \( \triangle ABC \) relate to fixed points and how the position of \( C \) changes with a variation in the angle \( \theta \). The concepts of sine, cosine, and especially tangent help set up the equations that represent the locus of point \( C \).

Angular Bisectors

Angular bisectors are lines or segments that divide an angle into two equal parts. In geometry, the concept of an angular bisector is significant because it helps in understanding the internal proportions and positioning of points relative to the angles.

When two angles differ by an angle \( \theta \) as in the problem involving \( \angle A \) and \( \angle B \), the angular bisector plays a crucial role. It defines how point \( C \) is positioned based on the differences in angles at \( A \) and \( B \).

The use of the angular bisector means that there is a unique line or path that point \( C \) follows as it relates to the divisors of \( \angle A \) and \( \angle B \). This establishes the geometric relationship necessary to derive the equation for the locus of \( C \). The bisector ensures that any change in angle \( \theta \) affects the locus path traced by point \( C \), as expressed by the tangent and cotangent in the solution.

Geometric Locus

The term 'geometric locus' refers to a set of points that satisfy particular geometric constraints or conditions. In simpler terms, it's the path or area occupied by a point, such as point \( C \) in this exercise, given these constraints.

In the context of triangles, the locus can be understood by examining the geometric properties governed by angle and distance. Specifically, for triangle \( \triangle ABC \), the locus of point \( C \) is a path that can be mathematically described once the relationships between the angles and sides are known.

Identifying the locus involves solving for \( C \)'s position based on the fixed points \( A \) and \( B \) and the given angle difference, \( \angle A - \angle B = \theta \). The locus equation, \( x^2 + y^2 + 2xy \tan \theta = a^2 \), effectively describes the geometric path that point \( C \) can occupy, under these defined conditions.
Ultimately, understanding the concept of locus helps in visualizing and solving geometric problems, aiding in determining where a point can exist relative to others in a system.