Problem 43 - Precalculus Mathematics for Calculus

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

Asymptotes are perpendicular (slopes: 1 and -1). Hyperbola with perpendicular asymptotes: \(x^2 - y^2 = \frac{c^2}{2}\).

Step 1: Recognize the General Equation

The standard form of a hyperbola centered at the origin with equation \(x^2 - y^2 = 5\) can be written as \(\frac{x^2}{5} - \frac{y^2}{5} = 1\). This identifies it as a hyperbola with horizontal transverse axis.

Step 2: Identify the Asymptotes

For the hyperbola in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equations of the asymptotes are given by \(y = \pm \frac{b}{a}x\). Here, \(a^2 = 5\) and \(b^2 = 5\). So, \(b/a = \sqrt{5}/\sqrt{5} = 1\). Thus, the asymptotes are \(y = x\) and \(y = -x\).

Step 3: Show Perpendicularity of Asymptotes

The slope of the asymptote \(y = x\) is 1, and the slope of \(y = -x\) is -1. Two lines are perpendicular if the product of their slopes is -1: \(1 \times (-1) = -1\). Thus, the asymptotes are perpendicular.

Step 4: Consider General Equation of Hyperbola

For asymptotes to be perpendicular, the hyperbola must be of the form \(x^2 - y^2 = k\), where \(k\) is a constant. This configuration ensures \(a = b\), resulting in perpendicular asymptotes with slopes \(\pm 1\).

Step 5: Find the Equation for the Hyperbola with Given Foci

Given foci \((\pm c, 0)\) implies that \(a^2 + b^2 = c^2\). With perpendicular asymptotes, we know \(a = b\), thus \(2a^2 = c^2\). Solving for \(a^2\) gives \(a^2 = c^2/2\). The equation is \(\frac{x^2}{c^2/2} - \frac{y^2}{c^2/2} = 1\) or \(x^2 - y^2 = c^2/2\).

Key Concepts

hyperbola equation

A hyperbola is a type of conic section that looks somewhat like two mirrored curves. These curves are formed when a plane cuts through both nappes (the upper and lower parts) of a double cone.
The general equation of a hyperbola with a horizontal transverse axis is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]In this form:

• \(a^2\) is associated with the positive term \(x^2\), indicating the direction of the transverse axis.
• \(b^2\) corresponds to the negative term \(y^2\), representing the conjugate axis.

For a vertical transverse axis, the equation is slightly different:
\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]The choice between these forms determines the orientation of the hyperbola along with its geometric properties.

asymptotes of hyperbola

Asymptotes are crucial lines that a hyperbola approaches but never touches. They act as invisible guidelines that shape the hyperbola's behavior from afar.
For a hyperbola in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equations for the asymptotes are:

• \(y = \frac{b}{a}x\)
• \(y = -\frac{b}{a}x\)

These asymptotes intersect at the hyperbola's center, providing a cross shape. If the hyperbola is shifted from the origin, the center needs to be factored into the equations of the asymptotes. In the fundamental scenario where \(a = b\), these asymptotes become \(y = x\) and \(y = -x\), indicating they are perpendicular, forming a right angle at the center of the hyperbola.

perpendicular lines

Perpendicular lines intersect at a right angle, which means they form a 90-degree angle with one another. In mathematical terms, two lines are perpendicular if the product of their slopes equals
\(-1\).

• If line 1 has a slope of \(m_1\) and line 2 has a slope of \(m_2\), then \(m_1 \times m_2 = -1 \).

In the context of hyperbolas, particularly the one described by \(x^2 - y^2 = 5\), the asymptotes \(y = x\) and \(y = -x\) are perpendicular. This is because their slopes, 1 and -1 respectively, multiply to
\(-1\).
Using perpendicular lines as asymptotes often suggests a symmetric and neatly oriented hyperbola, emphasizing its structure around the center.

hyperbola with foci at (±c, 0)

The foci of a hyperbola determine its precise shape and are located along its transverse axis. When given the foci at
\((\pm c, 0)\),
we know the foci rest horizontally on the x-axis.
The relationship between the foci and the hyperbola's dimensions is expressed by the equation:

• \(a^2 + b^2 = c^2\).

This equation ties the distance to the foci directly to the measurements of the hyperbola. In situations where the asymptotes must be perpendicular, it becomes clear that \(a = b\), simplifying our equation to
\(2a^2 = c^2\).
This relationship allows us to express the equation of the hyperbola succinctly as:

• \(x^2 - y^2 = c^2/2\)

Here, the symmetry of the hyperbola is emphasized by its equal axial stretch, and the positioning of its foci provides insight into its orientation and dispersion.