Problem 77 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Assertion: The combined equation of the asymptotes of the hyperbola \(2 x^{2}+5 x y+2 y^{2}+4 x+5 y+2=0\) Reason: The equation of a hyperbola and 1 ts asymptotes differ in constant terms only.

Step-by-Step Solution

Answer

Both Assertion and Reason are true. The Reason correctly explains the Assertion.

Step 1: Identify the equation

Hyperbola: \( 2x^2 + 5xy + 2y^2 + 4x + 5y + 2 = 0 \).

Step 2: Asymptote form

Asymptotes: \( 2x^2 + 5xy + 2y^2 + 4x + 5y + k = 0 \). For pair of lines, \( \Delta = 0 \).

Step 3: Compute determinant

\( \begin{vmatrix} 2 & 5/2 & 2 \\ 5/2 & 2 & 5/2 \\ 2 & 5/2 & k \end{vmatrix} = -\frac{9k}{4} + \frac{9}{2} = 0 \Rightarrow k = 2 \).

Step 4: Conclusion

Since \( k = F = 2 \), asymptotes equation equals hyperbola equation. Both Assertion and Reason are true; Reason explains Assertion.

Key Concepts

Asymptotes of Hyperbola

Asymptotes are crucial components of a hyperbola. They are straight lines that approach the hyperbola but never actually intersect it except possibly at infinity. Think of them as guides that the hyperbola follows closely, especially as the values of variables grow larger. These lines help in defining the overall shape of the hyperbola and are key in graphing it.

To find the asymptotes of a hyperbola represented by the general quadratic equation, we need the terms except the constant. Essentially, you set the constant term to zero and solve for the equation. In the provided exercise, we take:

• The given equation: \[2x^2 + 5xy + 2y^2 + 4x + 5y + 2 = 0\]
• Remove the constant term "2": \[2x^2 + 5xy + 2y^2 + 4x + 5y = 0\]

This equation represents the asymptotes of the original hyperbola equation. Asymptotes, therefore, serve as an approximate boundary which helps in visualizing and understanding the behavior of the hyperbola from a geometric perspective.

General Equation of Hyperbola

The general equation for a hyperbola can take different forms, but it often looks like this: \[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\] where each of the letters represents a coefficient that affects the shape and orientation of the hyperbola.

In our specific problem, the equation was given as: \[2x^2 + 5xy + 2y^2 + 4x + 5y + 2 = 0\] By comparing it to the general form, we note the coefficients:

• \(A = 2\), which is the coefficient for \(x^2\)
• \(B = 5\), which represents the coefficient of the mixed product term \(xy\)
• \(C = 2\), which is the coefficient for \(y^2\)
• \(D = 4\), related to the \(x\) term
• \(E = 5\), linked to the \(y\) term
• \(F = 2\), the constant

Identifying these coefficients is essential for transforming the equation into standard forms and aids in further understanding properties like the slopes of the asymptotes, center, and orientation of the hyperbola.

Mathematics Problem Solving

Problem-solving in mathematics involves a systematic approach to understanding and tackling issues by breaking them into manageable parts. In the context of the given problem, we start by identifying the known elements, such as the equation provided.

The first step involves recognizing that we have a quadratic equation in two variables. By comparing this with the general form \[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]we can identify corresponding coefficients. This helps in simplifying the problem as removing the constant term leads us directly to the equation of the asymptotes.

Such techniques of isolating terms, identifying key features, and recognizing patterns come under the broad umbrella of problem-solving methodologies. Practicing these techniques not only empowers students to solve the present problem but also builds a foundation for tackling more complex geometric challenges in the future. Understanding the fundamentals of problem-solving remains an invaluable skill across all areas of mathematics.