Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The conic is a vertical ellipse.

Step 1: Write the Equation in Standard Form

The given equation is \(9x^2 + 4y^2 = 9\). To write this equation in its standard form, divide every term by 9 to normalize it.The equation becomes:\[\frac{x^2}{1} + \frac{y^2}{\frac{9}{4}} = 1\] which simplifies to \[ \frac{x^2}{1} + \frac{y^2}{\left( \frac{3}{2} \right)^2} = 1 \].

Step 2: Identify the Type of Conic Section

The standard form equation \( \frac{x^2}{1} + \frac{y^2}{(\frac{3}{2})^2} = 1 \) fits the general form of the ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where both coefficients are positive and distinct.Here, \(a^2=1\) and \(b^2 = (\frac{3}{2})^2 = \frac{9}{4}\), with \(b^2 > a^2\), indicating that the major axis is vertical.

Key Concepts

Ellipse

An ellipse is a beautiful shape that is part of a family of curves known as conic sections. It appears like an elongated circle, commonly seen in planetary orbits and architectural structures. In mathematical terms, an ellipse can be defined as the set of all points for which the sum of the distances from two fixed points (the foci) is constant. This geometric figure can have a variety of orientations, either stretching more along the x-axis or the y-axis, classifying them as horizontal or vertical ellipses respectively.
Understanding ellipses is essential for grasping key concepts in geometry and calculus, as they often appear in various applications, including physics and engineering. The importance of recognizing an ellipse's equation allows us to further analyze the properties and dimensions, such as its axes and eccentricity. Generally, the beauty of an ellipse lies in its symmetry and the elegance of its curves, making it a fundamental element in the study of conic sections.

Standard Form of a Conic

The standard form of a conic section is instrumental for identifying and analyzing the type of conic you have. This standardized way to express the equation clarifies the geometrical interpretation and makes solving problems more manageable. Typically, for ellipses, the equation takes the form:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

Here, \(a\) and \(b\) are constants that represent the distances from the center to the ellipse's axes.
The standard form highlights the attributes of the conic, such as axes lengths and their orientation on the coordinate grid. For example, if \(b > a\), it implies the ellipse is taller rather than wider. Adjusting to the standard form means making comparisons easy with other conics, such as hyperbolas or parabolas. Users can apply common geometric properties and transformations confidently when everything is aligned to the standard form. Simplifying equations to this form unlocks ease of calculation and visualization.

Major Axis

The major axis of an ellipse is the longest diameter, stretching through the ellipse's center from one end to the other. Determining its length and direction is important for analyzing the ellipse's shape and orientation. As established from the standard form, the values \(a\) and \(b\) dictate whether an ellipse's major axis is horizontal or vertical.
If \(b > a\), as shown in the example \( \frac{x^2}{1} + \frac{y^2}{\left( \frac{3}{2} \right)^2} = 1 \), the major axis lies along the y-axis, making it vertical. In contrast, a larger \(a\) value would suggest a horizontal major axis.
Recognizing the major axis aids in comprehending the full dimensions and geometry of the ellipse. This axis acts as a backbone, around which the entire ellipse's geometry revolves, determining important features like the foci positions and length of the minor axis. Understanding these concepts is vital for accurately graphing and applying elliptical properties in numerous scientific and mathematical fields.