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

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) \(\frac{x}{y_{i}-y_{2}}+\frac{y}{x_{1}-x_{2}}=1\)

Step-by-Step Solution

Answer

The answer is option (A): \( \frac{x}{x_1 + x_2} + \frac{y}{y_1 + y_2} = 1 \).

Step 1: Substituting points into hyperbola equation

We start with the equation of the rectangular hyperbola \( xy = c^2 \). The points \((x_1, y_1)\) and \((x_2, y_2)\) lie on the hyperbola, hence \(x_1 y_1 = c^2\) and \(x_2 y_2 = c^2\).

Step 2: Deriving the chord equation

The equation of the chord joining these points can be determined from the condition that the general form of a line \((y - y_1) = m(x - x_1)\) must satisfy the hyperbola equation with both points \((x_1, y_1)\) and \((x_2, y_2)\).

Step 3: Slope calculation

The slope \( m \) of the line connecting the points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). We use this to form the line equation \(y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\).

Step 4: Using midpoint formula for hyperbola

For any chord of a hyperbola \( xy = c^2 \), the equation of the chord can also be rewritten by substituting \((x,y)\) with the midpoint values using \(x_c = \frac{x_1 + x_2}{2}\) and \(y_c = \frac{y_1 + y_2}{2}\), giving the chord equation \(x_c y + y_c x = x_c y_c\) directly using the fact both endpoints satisfy \(x_1 y_1 = x_2 y_2 = c^2\).

Step 5: Simplifying to match given options

Reorganizing the equation in Step 4, we can simplify to the form matching one of the provided options. After substitution and rearranging: \[ \frac{x}{x_1 + x_2} + \frac{y}{y_1 + y_2} = 1 \]. This equation matches option (A).

Key Concepts

Chord of Hyperbola

A chord of a hyperbola is essentially a straight line segment that joins two points on the curve of the hyperbola. In this context, we are specifically dealing with a rectangular hyperbola. This hyperbola has the standard form given by the equation: \[ xy = c^2 \] Here, \( c \) is a constant. When two points \((x_1, y_1)\) and \((x_2, y_2)\) are located on this hyperbola, they naturally satisfy the equation mentioned above, meaning \(x_1y_1 = c^2\) and \(x_2y_2 = c^2\).
The task is to find the precise equation that defines the chord connecting these two points. By substituting these points into the hyperbola’s equation, we ensure they lie on the hyperbola. This substitution is crucial in deriving the accurate equation for the chord, which explains how the chord aligns with the hyperbola's curvature.
This approach ultimately results in a chord equation that fits one of the given multiple-choice options in problems focused on hyperbolic geometry.

Midpoint Formula

The midpoint formula is a useful mathematical tool to find the central point between two coordinates in a plane. For any two points \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint \((x_c, y_c)\) is calculated using the formula:

• \( x_c = \frac{x_1 + x_2}{2} \)
• \( y_c = \frac{y_1 + y_2}{2} \)

This simplicity of the formula helps us determine exactly where the midpoint is located on the line segment between the two points.
In the context of a rectangular hyperbola, you can replace \((x, y)\) in the chord equation with their midpoint values. This substitution is significant because it aids the simplification of the equation, aligning it with the structure of the hyperbola itself.
Moreover, using midpoint values in hyperbolas ensures the consistent representation of curves and lines on these conic sections, providing a clear path towards establishing the full chord equation in harmony with the entire hyperbolic structure.

Equation of Line

The equation of a line is a fundamental concept in geometry and algebra. It's generally expressed in the form:\[ y - y_1 = m(x - x_1) \]where \(m\) is the slope, and \((x_1, y_1)\) is a particular point on the line. This form is often called the point-slope form.
When dealing with a hyperbola, particularly one that is rectangular, it's important to find the line equation that not only passes through points \((x_1, y_1)\) and \((x_2, y_2)\) but also integrates seamlessly with the hyperbolic structure.
The equation of the chord is effectively an expression of this line, ensuring that it connects two distinct points on the hyperbola, thus forming a complete line segment known as the chord. This equation describes both the geometric behavior and algebraic constraints the line must satisfy, crucial for defining and examining conic curves and their segments.

Slope Calculation

The slope of a line is an essential concept that measures its steepness and direction. When you have two points on a line, the slope \(m\) is calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This slope quantifies how much \(y\) changes for a certain change in \(x\), providing insight into the line's angle and orientation.
For the case of a line segment or chord on a rectangular hyperbola, the slope is key in constructing the line equation between points \((x_1, y_1)\) and \((x_2, y_2)\).

• It defines the alignment and trajectory of the chord.
• Enables the transformation of the point-slope form into the standard line equation.

This calculation allows us to seamlessly integrate the equation of the line with the properties of the hyperbola and find the exact solution aligning with the given problem context.Slope ensures the derived chord equation truly reflects the relationships and intersections dictated by the enhancing architectures of conic sections like hyperbolas.