Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the conic (horizontal ellipse, vertical hyperbola, and so on) corresponding to the given equation. $$ \frac{-x^{2}}{9}+\frac{y^{2}}{4}=1 $$

Step-by-Step Solution

Answer

The equation represents a vertical hyperbola.

Step 1: Identify the General Form

The given equation is \( \frac{-x^{2}}{9} + \frac{y^{2}}{4} = 1 \). Recognize that this resembles the general form of a hyperbola equation: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \).

Step 2: Determine the signs

Notice the equation has one negative term \( \frac{-x^{2}}{9} \) and one positive term \( \frac{y^{2}}{4} \). This indicates the equation is of a hyperbola since hyperbolas are characterized by one positive and one negative term when on opposite sides of the equation.

Step 3: Identify Hyperbola Orientation

Compare with the standard forms: for a horizontal hyperbola, it is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), and for a vertical hyperbola, it is \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \). Here, the positive term is \( \frac{y^2}{4} \), which matches the vertical hyperbola form \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \).

Step 4: Check the Denominators

Confirm that the denominators \( 9 \) and \( 4 \) provide the values for \( a^2 \) and \( b^2 \). For a vertical hyperbola, \( a^2 = 9 \) and \( b^2 = 4 \) indicate the vertical orientation as consistent with the typical hyperbola form.

Key Concepts

Hyperbola

A hyperbola is a type of conic section that forms an open curve. Imagine slicing a double cone (two cones placed tip-to-tip) at a certain angle, and you'll get a hyperbola. This geometric shape is distinct in having two separate curves, known as "branches," instead of being a closed shape like an ellipse or a circle. Hyperbolas are used in various fields, including physics and engineering. For example, they describe the paths of some comets and light beams. They also appear in radio waves and are useful in navigation systems. A hyperbola is defined by its equation, which involves two squared terms where one of them is negative. This negative square distinguishes it from other conic sections, like ellipses and circles. The general form of a hyperbola's equation is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \). These equations tell us important properties about the hyperbola, including the axes of symmetry and the orientation. Understanding hyperbolas involves analyzing the signs and positions of terms in their equations. This helps determine their specific orientation, whether they extend horizontally or vertically.

Vertical Hyperbola

A vertical hyperbola is a specific orientation of a hyperbola where the branches open up and down. It's like the familiar shape of a parabola extending in both the upward and downward directions.A vertical hyperbola's equation takes the form \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \). This arrangement signifies that the main focal direction is along the y-axis, and the term \( y^2 \) is positive. This is a significant identifier in distinguishing between vertical and horizontal hyperbolas. The constants \( a^2 \) and \( b^2 \) in the equation relate to the hyperbola's dimensions:

• \( a \) determines the distance from the center to the vertices along the x-axis.
• \( b \) determines the distance from the center to the points of the opening along the y-axis.

For our original equation example, \( \frac{y^2}{4} - \frac{x^2}{9} = 1 \), it is clear the hyperbola is vertical. The placement of the terms is crucial here; since the \( y^2 \) term is positive, it indicates a vertical orientation.

Equation of a Hyperbola

The equation of a hyperbola is a tool that helps describe its structure and orientation through mathematical expression. It generally involves two squared terms that are either added or subtracted. This results in an open curve pattern that makes the hyperbola distinct from other conic sections. The standard form of a hyperbola equation can appear in two ways:

• \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) – representing a horizontal hyperbola, which opens to the left and right.
• \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \) – representing a vertical hyperbola, which opens up and down.

In the equation, the letters \( a^2 \) and \( b^2 \) affect the hyperbola's dimensions. They are not equal to the lengths of the axes but they relate to these distances:

• \( a \) indicates distance from the center to vertex along the primary direction (x for horizontal, y for vertical).
• \( b \) describes distance from the center to the co-vertices, perpendicular to the primary direction.

By examining the original equation \( \frac{-x^{2}}{9} + \frac{y^{2}}{4} = 1 \), we recognize this is in the vertical hyperbola format. The placement of terms and the sign in front of \( x^2 \) demonstrate its vertical nature. This understanding comes from scrutinizing how the hyperbola's terms interact to form its unique shape.