Problem 51 - Contemporary Precalculus

Question

Show that the asymptotes of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{a^{2}}=1\) are perpendicular to each other.

Step-by-Step Solution

Answer

Answer: Yes, the asymptotes of the hyperbola are perpendicular to each other.

Step 1: Find the equations of the asymptotes

The general equation for a hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). The asymptotes of this hyperbola have the equations \(y = \pm \frac{b}{a}x\). Since the given hyperbola has its equation as \(\frac{x^2}{a^2} - \frac{y^2}{a^2} = 1\), its asymptotes are \(y = \pm \frac{a}{a}x\), which simplifies to \(y = \pm x\).

Step 2: Calculate the product of the slopes

The slopes of the asymptotes are the coefficients of \(x\) in their equations. The slope of the first asymptote is \(m_1 = 1\), and the slope of the second asymptote is \(m_2 = -1\). To check if they are perpendicular, compute the product of the slopes: \(m_1 \times m_2 = 1 \times -1 = -1\).

Step 3: Conclude if the asymptotes are perpendicular

Since the product of the slopes of the asymptotes is -1, it can be concluded that the asymptotes of the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{a^2} = 1\) are perpendicular to each other.

Key Concepts

Asymptotes

Asymptotes are invisible lines that a hyperbola extends closer to as it stretches toward infinity. They do not meet the hyperbola but give a helpful visual guide to its shape. Generally, for a hyperbola of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), we derive the equations of its asymptotes from the expressions \(y = \pm \frac{b}{a}x\). These lines reflect the direction the branches of the hyperbola extend. For instance, when \(a = b\) in the equation \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{a^{2}}=1\), the asymptotes simplify to \(y = \pm x\). This means each asymptote has an origin point and moves diagonally in opposite directions.

Perpendicular Slopes

The concept of perpendicular slopes is essential for recognizing the geometry of lines. Two lines are considered perpendicular if the product of their slopes equals -1. Slope represents how steep a line is, and it's often labeled \(m\) in equations. For hyperbolas like \(\frac{x^2}{a^2} - \frac{y^2}{a^2} = 1\), examining the slopes \(m_1\) and \(m_2\) of the simpler equations \(y = \pm x\) reveals their perpendicular nature. This is because \(m_1 = 1\) and \(m_2 = -1\), and their product \(1 \times -1 = -1\), showing conclusively they angle at 90 degrees to each other. This geometric property assures their perpendicularity, enhancing the understanding of hyperbola behavior.

Equations of Hyperbola

Understanding the equation of a hyperbola is key to grasping its shape and behavior. A hyperbola is one of the conic sections, and its standard equation can appear in two forms: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\). These alter whether the hyperbola opens along the x-axis or y-axis. An equation such as \(\frac{x^2}{a^2} - \frac{y^2}{a^2} = 1\) indicates equal stretching in both directions, framing a hyperbola with perpendicular asymptotes \(y = \pm x\).
Full comprehension of these equations involves recognizing constants \(a\) and \(b\), which guide the dimensions and orientation of the hyperbola. When \(a = b\), as in the mentioned hyperbola, symmetries simplify, making it easier to solve related geometric problems.