Problem 5 - Intermediate Algebra

Question

Identify the graph of each equation as an ellipse or a hyperbola. Do not graph. \(-\frac{y^{2}}{25}+\frac{x^{2}}{36}=1\)

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Recognize the Standard Forms

The standard form of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). The standard form of a hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( -\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). In our problem, the equation is \( -\frac{y^{2}}{25} + \frac{x^{2}}{36} = 1 \).

Step 2: Compare With Standard Form

Compare the given equation \( -\frac{y^{2}}{25} + \frac{x^{2}}{36} = 1 \) with the standard forms of conic sections. Notice that the equation has a negative term \(-\frac{y^2}{25}\).

Step 3: Identify the Type of Conic Section

Given the presence of one positive and one negative term and that the equation matches the standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the equation represents a hyperbola. The individual term for \( y^2 \) is subtracted in the equation, characteristic of the hyperbola form.

Key Concepts

Hyperbola

A hyperbola is a type of conic section formed by the intersection of a double cone with a plane. It is characterized by two distinct open curves, called branches, that face away from each other. The standard form of a hyperbola includes terms for both the horizontal and vertical axes, but crucially, one of these terms will be negative.

Here are the key elements of a hyperbola:

• The equation has one minus sign between the squared terms, indicating one of them is subtracted.
• The general form is either \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( -\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
• In a hyperbola equation, the positive variable indicates the axis along which the branches will open.

Hyperbolas have important characteristics such as asymptotes, which are imaginary lines that the graphs approach but never touch. These features make hyperbolas unique and useful in various fields like navigation and physics.

Ellipse

An ellipse is another type of conic section, often easily recognizable by its oval shape. Unlike hyperbolas with their distinct branches, ellipses are closed curves that look like elongated circles.

Key features of ellipses include:

• Both squared terms are added in the standard form.
• The standard form of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
• The axis with the larger denominator indicates the major axis, providing the direction in which the ellipse stretches more.

Ellipses are important in many mathematical models, especially in astronomy, where they describe the orbits of planets and moons. The symmetry and closed nature make ellipses well-suited for these applications. Elliptical shapes are also seen in various architectural designs due to their aesthetic appeal and structural characteristics.

Standard Forms of Conic Sections

Conic sections, fundamental in mathematics, are curves obtained by slicing through a cone at different angles. These sections include circles, ellipses, parabolas, and hyperbolas. Each type of conic section has a unique standard form based on its geometrical properties.

The standard forms for these sections are:

• Circle: \( x^2 + y^2 = r^2 \) where \( r \) is the radius.
• Ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
• Hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( -\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
• Parabola: \( y^2 = 4ax \) or \( x^2 = 4ay \)

Understanding these properties lets us identify and graph conic sections easily in algebra and geometry. Grasping these equations' differences helps students and professionals apply them in real-world scenarios, such as engineering, astronomy, and physics.