Problem 36 - Thomas Calculus

Question

Exercises \(35-38\) give foci and corresponding directrices of hyperbolas centered at the origin of the \(x y\) -plane. In each case, find the hyperbola's eccentricity. Then find the hyperbola's standard-form equation. $$ \begin{array}{l}{\text { Focus: }(\sqrt{10}, 0)} \\ {\text { Directrix: } \quad x=\sqrt{2}}\end{array} $$

Step-by-Step Solution

Answer

Eccentricity is \(\sqrt{5}\); Equation: \(\frac{x^2}{2} - \frac{y^2}{8} = 1\).

Step 1: Understanding Hyperbola Properties

A hyperbola is defined in terms of its two foci and its directrix. Here, we are given a focus at \((\sqrt{10}, 0)\) and a directrix located at \(x = \sqrt{2}\). These properties are crucial in defining the hyperbola's eccentricity and equation.

Step 2: Define Eccentricity

Eccentricity \(e\) of a hyperbola is defined as \(e = \frac{c}{a}\), where \(c\) is the distance from the center to a focus, and \(a\) is the distance from the center to the directrix. We know \(c = \sqrt{10}\) and the directrix tells us \(a = \sqrt{2}\).

Step 3: Calculate Eccentricity

To find the eccentricity \(e\), use the eccentricity formula: \(e = \frac{c}{a}\). Substitute \(c = \sqrt{10}\) and \(a = \sqrt{2}\) to get \(e = \frac{\sqrt{10}}{\sqrt{2}} = \sqrt{\frac{10}{2}} = \sqrt{5}\).

Step 4: Standard Form Equation of Hyperbola

The standard form for a hyperbola centered at the origin with a horizontal transverse axis is given by \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Since \(a = \sqrt{2}\), we have \(a^2 = 2\). Use the relation \(c^2 = a^2 + b^2\) to find \(b^2\): \(10 = 2 + b^2\), so \(b^2 = 8\).

Step 5: Substitute and Simplify

Substitute \(a^2 = 2\) and \(b^2 = 8\) into the standard form: \(\frac{x^2}{2} - \frac{y^2}{8} = 1\). This is the hyperbola's equation in standard form.

Key Concepts

Eccentricity

The concept of eccentricity is pivotal in understanding the shape and properties of a hyperbola. In simple terms, eccentricity (denoted as \( e \)) measures how much a conic section deviates from being circular. For hyperbolas, it is always greater than 1.

Eccentricity is calculated using the formula \( e = \frac{c}{a} \), where:

• \( c \) is the distance from the center to one of the foci.
• \( a \) is the distance from the center to the corresponding directrix.

In our given example, with a focus at \((\sqrt{10}, 0)\) and the directrix at \(x = \sqrt{2}\), we find \( c = \sqrt{10} \) and \( a = \sqrt{2} \).

• Substituting these values into the formula gives \( e = \frac{\sqrt{10}}{\sqrt{2}} = \sqrt{5} \).

This value of \( e \) indicates how stretched the hyperbola is compared to a circle.

Foci of Hyperbola

The foci (plural of focus) are two distinct points used in defining a hyperbola. These points play a key role in characterizing the shape of the hyperbola and determining its equation.

The main property of a hyperbola is that the absolute difference in distances from any point on the hyperbola to the two foci is constant. In our problem, the coordinates of one focus given are \((\sqrt{10}, 0)\). For hyperbolas centered at the origin, the foci are symmetrically placed along the transverse axis.
For this hyperbola with a horizontal transverse axis, foci are \((\pm \sqrt{10}, 0)\):

• They lie on the x-axis because the transverse axis is horizontal.
• These points are crucial in both constructing and understanding the geometry of hyperbolas.

Directrix

The directrix is a fixed line used alongside the focus to define a hyperbola. Each hyperbola is associated with a pair of directrices that lie outside the curve.

In our example, the directrix is given by the equation \(x = \sqrt{2}\). This means:

• The directrix is parallel to the y-axis.
• It determines how stretched or compressed the hyperbola is, relative to the foci.

The concept of directrix helps in setting up the ratio used to calculate eccentricity. The distance from the center to a directrix \((x = \sqrt{2})\) is equal to \( a = \sqrt{2} \). This information combined with the formula for eccentricity helps us solve the equation of the hyperbola.

Standard Form of Hyperbola

Finding the standard form of a hyperbola is an essential step in visualizing the curve. For hyperbolas centered at the origin with a horizontal transverse axis, the standard form of the equation is:\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]

Given \( a = \sqrt{2} \), we find \( a^2 = 2 \). Now use the hyperbola relation \( c^2 = a^2 + b^2 \) to calculate \( b^2 \). With \( c = \sqrt{10} \), the equation \( 10 = 2 + b^2 \) helps us find:

• \( b^2 = 8 \)

Substituting back into the standard form results in the final equation:\[\frac{x^2}{2} - \frac{y^2}{8} = 1\]
This equation describes the hyperbola in terms of its horizontal stretch and vertical compression, encapsulating its unique geometric properties.