Problem 1 - 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

Horizontal ellipse.

Step 1: Identify the Given Equation Form

The given equation is \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \). This equation is not explicitly in the standard form. However, we can compare it to conic section forms to identify what type of conic it represents.

Step 2: Recall the Standard Form of an Ellipse

In standard form, the equation of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) or \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \) where \(a^2\) and \(b^2\) are the denominators and represent the squares of the semi-major and semi-minor axes. Depending on whether \(a^2 > b^2\) or \(b^2 > a^2\), the major axis is either horizontal or vertical.

Step 3: Compare with the Standard Ellipse Equation

The given equation \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \) matches the standard form of \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \): here, \( a^2 = 9 \) and \( b^2 = 4 \). This means \( a = 3 \) and \( b = 2 \). Since \( a > b \), the ellipse is horizontal.

Key Concepts

Ellipse Identification

Identifying an ellipse from a mathematical equation involves recognizing its distinct form. The equation in the exercise is \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \). This is a perfect example of how ellipses are represented in coordinate geometry. When you see an equation of the type \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), you're dealing with an ellipse. If the sum of two squares is set equal to one (1), and both terms are positive, it's a tell-tale sign of an ellipse.

• If two terms are positive: It's an ellipse or a circle.
• The denominators determine axis lengths: Larger denominator indicates major axis direction.

In this case, the presence of \( x^2 \) and \( y^2 \) with positive coefficients, and both under fractions pointing to an equal sum of one, confirms the equation is for an ellipse.

Standard Form of Ellipse

Understanding the standard form of an ellipse is crucial for recognizing different characteristics of the ellipse at a glance. The general expression is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) or \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \). This tells us everything about the ellipse's orientation and size.It's like a recipe, where:

• \(a^2\) and \(b^2\) are ingredients that tell you the size of the ellipse's axes.
• Position of \(a^2\) and \(b^2\) determines the direction (horizontal or vertical).

By analyzing \( a^2 = 9 \) and \( b^2 = 4 \), it becomes clear that the ellipse extends further along the direction of \( x \) as \( a > b \). This gives the ellipse its horizontal orientation. The equation gives us both the size of the axes and the shape of the ellipse, just through its arrangement and values.

Major and Minor Axes

The major and minor axes define the size and orientation of the ellipse.- **Major Axis:** This is the longest diameter of the ellipse. It passes through the center and the widest point of the ellipse. - **Minor Axis:** This is perpendicular to the major axis and is the shortest diameter.In our example, \( a^2 = 9 \) corresponds to the major axis, while \( b^2 = 4 \) is for the minor axis. Calculating the lengths:- The major axis, coming from \( a = 3 \), is 6 units long, stretching along the x-axis.- The minor axis, from \( b = 2 \), is 4 units long, aligned along the y-axis.Why are these important? Understanding these axes helps visualize how stretched or compact the ellipse is, and informs you about its orientation in the coordinate plane. Whether the ellipse is wider or taller depends on which denominator is larger, \( a^2 \) or \( b^2 \). This, in turn, affects any calculations about the ellipse, such as its circumference or area.