Problem 82 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equation of the chord joining two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) on the rectangular hyperbola \(x y=c^{2}\) is : (A) \(\frac{x}{x_{1}+x_{2}}+\frac{y}{y_{1}+y_{2}}=1\) (B) \(\frac{x}{x_{1}-x_{2}}+\frac{y}{y_{1}-y_{2}}=1\) (C) \(\frac{x}{y_{1}+y_{2}}+\frac{y}{x_{1}+x_{2}}=1\) (D) \(\frac{x}{y_{1}-y_{2}}+\frac{y}{x_{1}-x_{2}}=1\)

Step-by-Step Solution

Answer

Option C: \( \frac{x}{y_1 + y_2} + \frac{y}{x_1 + x_2} = 1 \).

Step 1: Identify the Equation of Hyperbola

The equation provided is for a rectangular hyperbola, which takes the form \( x y = c^2 \).

Step 2: Recall the Equation of a Chord on the Hyperbola

For a hyperbola \( x y = c^2 \), the equation of a chord between any two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on the hyperbola can be written using the section formula: \( x(y_1 + y_2) + y(x_1 + x_2) = 2c^2 \).

Step 3: Manipulate the Chord Equation

From the equation \( x(y_1 + y_2) + y(x_1 + x_2) = 2c^2 \), divide through by \( 2c^2 \) to express it in a different form. This gives: \( \frac{x(y_1 + y_2)}{2c^2} + \frac{y(x_1 + x_2)}{2c^2} = 1 \).

Step 4: Compare the Rewritten Equation with Given Options

Rewrite the expression \( \frac{x}{y_1 + y_2} + \frac{y}{x_1 + x_2} = 1\) which matches option C. The structure of this rewritten equation directly corresponds to the given option (C).

Key Concepts

Rectangular Hyperbola

A rectangular hyperbola is a special type of hyperbola. It is characterized by its symmetry around the coordinate axes. In the coordinate plane, a rectangular hyperbola can be defined by the equation \( xy = c^2 \), where \( c \) is a constant. This equation suggests that as one coordinate increases, the other must decrease to maintain the same product \( c^2 \).
This type of hyperbola has its asymptotes perpendicular to each other, intersecting at coordinates \( (0,0) \) on the graph. These asymptotes, \( x = 0 \) and \( y = 0 \), are essential in determining the hyperbola's shape and orientation.
Understanding the behavior of a rectangular hyperbola helps in solving related problems in coordinate geometry. It allows for predictions of how the graph behaves in any quadrant based on the sign and value of \( c \).

• Rectangular hyperbolas are symmetrical along both coordinate axes.
• The graph passes through the origin \( (0,0) \).
• The key characteristic is its constant value product \( xy = c^2 \).

Chord Equation

In the context of a hyperbola, a chord is a line segment that connects two points on the hyperbola. Understanding how to derive the equation of the chord is crucial for tackling complex coordinate geometry problems. On the hyperbola \( xy = c^2 \), the equation of a chord through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be written using the formula:
\[x(y_1 + y_2) + y(x_1 + x_2) = 2c^2\]
This equation can be understood as a linear relation between \( x \) and \( y \) and depicts how both coordinate components interact to conform with the hyperbolic condition.
The form of the chord equation is symmetric, reflecting the inherent symmetry of the hyperbola. It also highlights how both points contribute equally to the equation, a fundamental concept in geometry concerning lines and curves.
By dividing both sides by \( 2c^2 \), we simplify the equation to match typical answer choices in exams or textbook problems, helping compare and identify correct solutions swiftly.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, connects algebra with geometry. It provides a way to algebraically represent geometric figures using coordinates and equations. This branch of mathematics allows us to explore and solve problems involving lines, curves, and shapes on the Cartesian plane.
In coordinate geometry, the position of any point is defined using two numbers, \( x \) and \( y \) coordinates. Lines, curves, and shapes can then be expressed through equations, making it possible to analyze intersections, distances, angles, and areas.
When working with elements like the rectangular hyperbola or the chord equation, coordinate geometry principles become highly beneficial. They assist in determining precise locations and relationships between geometric entities on a plane.

• Converts geometric problems into algebraic format for easier manipulation.
• Allows precise calculations of distances, midpoints, and angles.
• Facilitates representation and exploration of complex shapes and their properties.

Coordinate geometry provides the foundational tools needed to translate the visual world into mathematical forms, making it an essential concept in mathematics education and applications.