Problem 77

Question

In the following questions an Assertion $(A)$ is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason( $\mathrm{R}$ ) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True 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 its asymptotes differ in constant terms only.

Step-by-Step Solution

Verified

Answer

(A) Both assertion and reason are true; reason explains the assertion.

1Step 1: Identify the Problem

The problem provides an assertion about the combined equation of the asymptotes of a hyperbola and a reason explaining properties of hyperbolas and their asymptotes. We need to verify both and determine the relationship between them.

2Step 2: Analyze the Assertion

The assertion states that the combined equation of the asymptotes of the hyperbola is given by the conic equation: \[ 2x^2 + 5xy + 2y^2 + 4x + 5y + 2 = 0. \] To find the combined asymptotes, we focus on the terms "$2x^2 + 5xy + 2y^2$" and ignore the others. Setting:\[ 2x^2 + 5xy + 2y^2 = 0, \]we obtain the combined equation of the asymptotes.

3Step 3: Verify the Reason

The reason states that the difference between the equation of a hyperbola and the equation of its asymptotes is in the constant term only. For the given hyperbola, the asymptotes are found by considering only the second degree terms and dropping all other terms, which means they do indeed differ only in the constant.

4Step 4: Determine the Relationship

Both the assertion and the reason are true. Since the assertion can be directly obtained by ignoring the constants in the general form of the hyperbola's equation, the reason correctly justifies the assertion as it directly explains how the constant term is treated in deriving the asymptotes.

Key Concepts

Understanding Asymptotes in HyperbolasConic Sections and HyperbolasThe Equation of Hyperbola and Its Asymptotes

Understanding Asymptotes in Hyperbolas

Asymptotes are lines that a curve approaches as it heads towards infinity. For hyperbolas, asymptotes provide a sort of "guiding-lines," helping us to understand the overall shape of the curve. A hyperbola has two asymptotes, which intersect at the hyperbola's center. These asymptotes define the directions in which the hyperbola will extend without ever actually touching these lines.

Asymptotes give you an idea of how wide a hyperbola opens.
They are crucial for sketching hyperbolas.

For a given hyperbola, its asymptotes can be found by eliminating the constant term from its equation. This process leaves you with the second-degree terms only. These terms form the combined equation of the asymptotes, capturing how the hyperbola's curve expands in both directions.
It's similar to peeling an onion, where every layer uncovers another, leading us finally to the core — the asymptotes.

Conic Sections and Hyperbolas

Conic sections are curves obtained by slicing a cone with a plane in different ways. These sections include circles, ellipses, parabolas, and hyperbolas. Hyperbolas occur when the plane makes a steep enough angle, slicing through both halves of the cone. Unlike a circle or ellipse, which is a closed curve, a hyperbola is an open curve consisting of two separate branches.

Conic sections have unique properties that distinguish each shape.
They all can be expressed using general quadratic equations.

Hyperbolas are represented by equations of the form: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]These involve both the square terms $x^2$ and $y^2$. The coefficient of the $xy$ term also plays a role in determining the hyperbola's orientation. When analyzed, these terms help define whether a conic is a hyperbola and where its asymptotes lie.

The Equation of Hyperbola and Its Asymptotes

The equation for a hyperbola gives vital information about its shape, size, and position. Primarily, we use the equation to find its vertex, center, and the asymptotes. By transforming or manipulating the equation, we can derive the asymptote equations, which are essential for understanding the hyperbola's geometric parameters.

Start with the general equation of a hyperbola.
Focus on the second-degree terms for asymptote equations.
Asymptotes delineate the boundary towards which the hyperbola opens.

To find the equation of a hyperbola's asymptotes, you set the linear and constant terms to zero, leaving only the quadratic terms. For example, in the equation $2x^2 + 5xy + 2y^2 + 4x + 5y + 2 = 0$, remove $4x + 5y + 2$, getting $2x^2 + 5xy + 2y^2 = 0$. This is the sought-after combined equation that defines the asymptotes of the hyperbola.
This small adjustment offers a deeper peek into the hyperbola’s enduring structure.

Problem 75

Problem 78

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