Problem 16 Identify the center of each elli... [FREE SOLUTION]

Chapter 14: Problem 16

Identify the center of each ellipse and graph the equation. $$\frac{x^{2}}{9}+y^{2}=1$$

Short Answer

Expert verified

The center of the ellipse is at the origin (0,0) with a semi-major axis of 3 units and a semi-minor axis of 1 unit. To graph the ellipse, mark the center (0,0) and draw points at (3,0), (-3,0), (0,1), and (0,-1). Then, connect these points with a smooth curve to form the ellipse.

Step by step solution

From the given ellipse equation, we see that: $h = 0$, $k = 0$, $a^2 = 9$, and $b^2 = 1$ So, the center of the ellipse is at the origin, (0,0). Now, let's find the semi-major and semi-minor axes: $a = \sqrt{9} = 3$ $b = \sqrt{1} = 1$ The semi-major axis (a) is 3 units, and the semi-minor axis (b) is 1 unit. #Step 2: Graph the ellipse#

Since the center of the ellipse is at the origin (0,0), the major axis is horizontal, and the minor axis is vertical. To graph the ellipse, follow these steps: 1. Mark the center of the ellipse (0,0) on the graph. 2. Move 3 units to the right from the center, and draw a point. This is one end of the major axis (3,0). 3. Move 3 units to the left from the center, and draw a point. This is the other end of the major axis (-3,0). 4. Move 1 unit up from the center, and draw a point. This is one end of the minor axis (0,1). 5. Move 1 unit down from the center, and draw a point. This is the other end of the minor axis (0,-1). 6. Connect the points you drew with a smooth curve to form the ellipse. The ellipse should now be accurately graphed with a center at the origin, horizontal major axis of length 6 units (2 * a), and a vertical minor axis of length 2 units (2 * b).

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Ellipses

Graphing an ellipse involves understanding its equation and translating it onto a coordinate plane. The standard form of an ellipse's equation is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$. Here, $(h, k)$ represents the center, while $a$ and $b$ denote the lengths of the semi-major and semi-minor axes, respectively.
To graph an ellipse:

Identify the center, $(h, k)$ from the equation.
Determine the values of $a$ and $b$ by taking the square roots of $a^2$ and $b^2$.
Use these values to plot points along the major and minor axes.
Connect these points smoothly to form the ellipse.

Remember, the major axis is along the direction of the larger number between $a$ and $b$.

Semi-Major Axis

The semi-major axis is the longest radius of an ellipse, extending from the center to the ellipse's edge. It lies along the axis where the denominator of the equation is larger.
For example, in the equation $\frac{x^{2}}{9}+\frac{y^{2}}{1}=1$, since $a^2 = 9$, the semi-major axis is 3, which is longer than the semi-minor axis. Key points:

It determines the direction and length of the ellipse's longest stretch.
The value $a$ is calculated by $a = \sqrt{a^2}$.
On the graph, the total length is $2a$, as it stretches both ways from the center.

This axis is crucial for accurately drawing the ellipse's longest dimensions.

Semi-Minor Axis

The semi-minor axis is the shortest radius of an ellipse, perpendicular to the semi-major axis. It helps define how "squished" the ellipse appears.
In the equation $\frac{x^{2}}{9}+\frac{y^{2}}{1}=1$, with $b^2=1$, the semi-minor axis is 1.Important aspects:

It provides the shortest diameter of the ellipse.
Its length is found using $b = \sqrt{b^2}$.
On the graph, the total length is $2b$.

Understanding both semi-major and semi-minor axes is essential in graphing ellipses effectively.

Conic Sections

Ellipses are a type of conic section, which are curves obtained by intersecting a plane with a cone. The four basic types of conic sections are circles, ellipses, parabolas, and hyperbolas.
Ellipses specifically occur when the intersecting plane cuts through both nappes of the cone but not as steep as parabolas or hyperbolas. Characteristics of ellipses:

They have two focal points that describe their shape.
Ellipses can become circles if the semi-major and semi-minor axes are equal.
Their unique properties and shapes have vital applications in astronomy, engineering, and design.

Studying conic sections helps in understanding the complex relationships and shapes within mathematics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Identify the center of each ellipse and graph the equation. $$\frac{x^{2}}{9}+y^{2}=1$$

Short Answer

Step by step solution

Key Concepts

Graphing Ellipses

Semi-Major Axis

Semi-Minor Axis

Conic Sections

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Mechanics Maths

Logic and Functions

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.