Problem 67 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The asymptotes of the hyperbola \(x y-3 x+4 y+2=0\) are (A) \(x=-4\) (B) \(x=4\) (C) \(y=-3\) (D) \(y=3\)

Step-by-Step Solution

Answer

Asymptotes are \( y=3 \) and \( x=-4 \).

Step 1: Identify the hyperbola equation

The given equation of the hyperbola is \( xy - 3x + 4y + 2 = 0 \). Our goal is to find the asymptotes of this hyperbola.

Step 2: Rearrange the equation

Rearrange the terms to group x-related and y-related terms: \( xy - 3x + 4y + 2 = 0 \) can be rewritten as \( xy - 3x + 4y = -2 \).

Step 3: Factor common terms

Factor the equation to isolate terms with x and y: \[ x(y - 3) + 4y = -2 \].

Step 4: Set equation to zero for asymptotes

The asymptotes of a hyperbola represented by a general quadratic equation can be found by setting the constant term to zero: \[ x(y - 3) + 4y = 0 \].

Step 5: Solve for x

Solve for \( x \) in terms of \( y \): \[ x(y-3) = -4y \] => \[ x = \frac{-4y}{y-3} \].

Step 6: Determine vertical asymptote

For vertical asymptotes, the denominator must be zero, thus \( y-3 = 0 \) implies \( y = 3 \). This represents the vertical asymptote.

Step 7: Solve for y

Conversely, solve for \( y \) in terms of \( x \): Rearranging gives \[ y(x+4) = 3x \], thus \[ y = \frac{3x}{x+4} \].

Step 8: Determine horizontal asymptote

For horizontal asymptotes, the denominator must be zero: \( x+4 = 0 \) yields \( x = -4 \), representing the horizontal asymptote.

Key Concepts

Quadratic Equations

Quadratic equations are equations of the form \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). These equations are named for the quadratic term \( ax^2 \), which is what gives the equation its parabolic graph. However, when dealing with hyperbolas, we often come across quadratic forms that involve both \( x \) and \( y \), such as \[ xy - 3x + 4y + 2 = 0. \] This particular form involves products of different variables, making it part of a different conic section analysis. Understanding quadratic equations will help in identifying how to rearrange and factor terms effectively to find asymptotes.

In the context of a hyperbola, the goal is to identify expressions that factor in such a way that resemble the core quadratic-shaped components within the equation. Often, hyperbolas involve cross-product terms like \( xy \) that require grouping and factoring to reveal asymptotic behavior.

Vertical Asymptote

A vertical asymptote is a vertical line that a graph approaches but never touches or crosses. For hyperbolas, finding a vertical asymptote involves determining the values of \( y \) that cause certain terms to be undefined or lead to a denominator of zero in rational expressions. In simpler terms, we investigate the hyperbola equation after factoring and setting up expressions that lead to a division by zero for the given variable.

In the equation \[ xy - 3x + 4y + 2 = 0, \]after rearranging and factoring, we find \[ x(y-3) + 4y = 0. \]The vertical asymptote arises when the term \( y-3 \) in \[ x = \frac{-4y}{y-3} \] causes a division by zero. Setting \( y-3 = 0 \), we uncover \( y = 3 \) as the vertical asymptote.

Finding vertical asymptotes is critical in understanding where the function increases or decreases without bound as it never intersects these lines.

Horizontal Asymptote

Horizontal asymptotes are lines that graphs approach as the independent variable tends to positive or negative infinity. For hyperbolas, identification of horizontal asymptotes involves rearranging the hyperbola's equation to solve for \( y \) in terms of \( x \) and finding values of \( x \) that cause the equation to be undefined or lead to a denominator of zero in rational expressions.

Considering the hyperbola equation \[ xy - 3x + 4y + 2 = 0, \]we rearranged and solved for \( y \) to get \[ y = \frac{3x}{x+4} .\]By inspecting the expression, we identify that \( x+4 = 0 \) when \( x = -4 \) gives rise to a horizontal asymptote. This is because if \( x+4 \) is zero, the entire expression for \( y \) becomes undefined.

Understanding horizontal asymptotes helps predict the behavior of a curve as it approaches infinity along the x-axis, providing insights into its end behavior.