Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Find the eccentricity e of each ellipse or hyperbola. $$x^{2}-y^{2}=4$$

Step-by-Step Solution

Answer

The eccentricity of the hyperbola is \( \sqrt{2} \).

Step 1: Identify the Conic Section

The given equation is \( x^{2} - y^{2} = 4 \). Since the equation is of the form \( x^{2} - y^{2} = c^{2} \), it represents a hyperbola, not an ellipse.

Step 2: Write the Standard Form of the Hyperbola

The standard form of a horizontal hyperbola centered at the origin is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). We can rewrite the given equation as \( \frac{x^2}{4} - \frac{y^2}{4} = 1 \) by dividing both sides by 4.

Step 3: Identify the Values for a and b

From the equation \( \frac{x^2}{4} - \frac{y^2}{4} = 1 \), we identify that \( a^2 = 4 \) and \( b^2 = 4 \). Thus, \( a = 2 \) and \( b = 2 \).

Step 4: Calculate the Eccentricity

For a hyperbola, the eccentricity \( e \) is calculated using the formula \( e = \sqrt{1 + \frac{b^2}{a^2}} \). We substitute \( a = 2 \) and \( b = 2 \) into the formula: \[ e = \sqrt{1 + \frac{4}{4}} = \sqrt{2} \].

Step 5: Conclusion

The eccentricity of the hyperbola is \( \sqrt{2} \).

Key Concepts

Hyperbola

A hyperbola is a type of conic section that is formed by intersecting a double cone with a plane in such a way that the angle between the plane and the axis of the cone is less than the angle between the generator of the cone and the axis. A hyperbola consists of two separate, symmetric curves known as branches. Each branch resembles an open curve. This separation is a defining characteristic of hyperbolas and distinguishes them from other conic sections, like circles and ellipses.

• Hyperbolas have two foci and two directrices, which are aligned along the transverse and conjugate axes.
• The standard position of a hyperbola, like other conic sections, is centered at the origin of a coordinate system.

Using these geometric properties, hyperbolas can be easily recognized within equations. A key sign of a hyperbola is when terms for both axes are squared with opposite signs in the equation, for example, the standard form equation includes the subtraction term such as \(x^2 - y^2\).
Understanding hyperbolas is crucial for tackling problems involving this conic section since many real-world applications, such as orbits and waves, can be modeled using hyperbolas.

Eccentricity

Eccentricity is a parameter that determines the shape of conic sections, including circles, ellipses, parabolas, and hyperbolas. For hyperbolas, the eccentricity, denoted by \(e\), is always greater than 1, which distinguishes it as a hyperbola among conic sections. This metric essentially measures how stretched or elongated a conic section is.

• The formula to calculate the eccentricity of a hyperbola is \(e = \sqrt{1 + \frac{b^2}{a^2}}\).
• In hyperbolas, the foci are located further from the center relative to the vertices, which is a reflection of their high eccentricity compared to ellipses, which have eccentricities less than 1.

The eccentricity concept helps in understanding not only the quantitative aspects of a hyperbola but also provides insights into its qualitative properties, like its openness and the degree of 'flattening' it represents compared to a circle or ellipse.

Standard Form of Hyperbola

The standard form of a hyperbola's equation is fundamental to identifying and solving problems related to these conic sections. For a hyperbola centered at the origin, the standard forms are:

For a horizontal hyperbola: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] For a vertical hyperbola: \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]

• The terms \(a^2\) and \(b^2\) represent squared lengths from the center to the vertices and co-vertices, respectively.
• The transverse axis lies along the axis with the positive fraction (either horizontal or vertical), dictating the direction of the hyperbola's opening.

In this context, converting a hyperbola's equation into its standard form is crucial for analysis, as it allows for straightforward calculation of key properties such as vertices, foci, and eccentricity. By dividing the entire equation by a common factor to match the form \(1\), you can compare it directly to its standard forms, making the identification of \(a\) and \(b\) possible.