Problem 39 - Algebra and Trigonometry

Question

(a) Show that the asymptotes of the hyperbola \(x^{2}-y^{2}=5\) are perpendicular to each other. (b) Find an equation for the hyperbola with foci \(( \pm c, 0)\) and with asymptotes perpendicular to each other.

Step-by-Step Solution

Answer

(a) Asymptotes: \(x = \pm y\). (b) Hyperbola: \(( x^2 - y^2 = a^2 )\) with \(a = c\).

Step 1: Understanding the Hyperbola Equation

The equation of a hyperbola in the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is given, where \( a^2 = b^2 \) results in an equation with perpendicular asymptotes. For \( x^2 - y^2 = 5 \), rewrite the equation as \( \frac{x^2}{5} - \frac{y^2}{5} = 1 \), so \( a^2 = b^2 = 5 \).

Step 2: Derive Asymptote Equations

For the hyperbola \( x^2 - y^2 = 5 \), the asymptotes are determined by setting the right side of \( \frac{x^2}{5} - \frac{y^2}{5} = 1 \) to zero: \( \frac{x^2}{5} - \frac{y^2}{5} = 0 \), leading to \( x^2 = y^2 \). Simplifying gives \( x = \pm y \). These asymptotes are perpendicular because the slopes, \( 1 \) and \( -1 \), are negative reciprocals of each other.

Step 3: Equation of Hyperbola with Given Foci and Perpendicular Asymptotes

When asymptotes are perpendicular, \( a^2 = b^2 \). Given the foci \( ( \pm c, 0) \), the equation of the hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{a^2} = 1 \), which simplifies to \( \frac{x^2 - y^2}{a^2} = 1 \). Since the asymptotes are perpendicular, the asymptote equation is \( x = \pm y \).

Key Concepts

Asymptotes

Asymptotes are straight lines that a curve approaches as it heads towards infinity. In the context of hyperbolas, asymptotes are important because they help define the shape and direction of the hyperbola's branches.
For a hyperbola represented by the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the asymptotes can be found by setting the right-hand side to zero: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 \).
This results in the equations \( x = \pm \frac{b}{a}y \), which tells us the direction of the asymptotes. These lines give us a clear boundary that the hyperbola will never meet, but will infinitely approach. Understanding asymptotes is crucial for analyzing and graphing hyperbolas in their simplest forms.

Perpendicular Asymptotes

In certain hyperbolas, the asymptotes are perpendicular to each other. This means that the angle between the asymptotes is 90 degrees. This occurs when \( a^2 = b^2 \).
In simpler terms, this results in the slopes of the asymptotes being negative reciprocals, such as \( 1 \) and \( -1 \). For example, consider the hyperbola \( x^2 - y^2 = 5 \).
When rewritten as \( \frac{x^2}{5} - \frac{y^2}{5} = 1 \), it's clear that here \( a^2 = b^2 = 5 \), confirming the asymptotes are indeed perpendicular. This symmetry in the hyperbola simplifies the analysis and calculation of its equations and intersections.

Equation of Hyperbola

The equation of a hyperbola gives us insight into its shape and properties. Hyperbolas are unique among conic sections because they consist of two symmetrical branches.
The standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is useful in identifying key features such as vertices and foci. However, for hyperbolas with perpendicular asymptotes, the equation simplifies to \( \frac{x^2 - y^2}{a^2} = 1 \).
This is derived when the asymptotes \( x = \pm y \) dictate that \( a^2 = b^2 \), resulting in beautifully symmetric branches. Understanding this form is vital, especially when determining the graph's orientation using its foci at \( ( \pm c, 0) \). This mathematical structure reveals much about the hyperbola's behavior and guiding lines.