/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Graph each of the following equa... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 24

Graph each of the following equations. $$ 9 y^{2}=9-x^{2} $$

Short Answer

Expert verified
The equation represents an ellipse centered at the origin with a vertical major axis of length 2 and a horizontal minor axis of length 6.

Step by step solution

01

Rewrite the equation

Rewrite the given equation into a more recognizable form. Start from the given equation: $$9 y^{2} = 9 - x^{2}$$Divide both sides by 9 to simplify: $$y^{2} = 1 - \frac{x^{2}}{9}$$
02

Recognize the standard form

Recognize that the equation is now in the form of $$y^2 = 1 - \frac{x^2}{9}$$, which can be rewritten as $$\frac{y^2}{1} + \frac{x^2}{9} = 1$$ This is the standard form of the equation of an ellipse centered at the origin, but with its axes swapped.
03

Identify key characteristics of the ellipse

Identify the key characteristics of the ellipse. For an ellipse in the form $$\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1$$, a = 1 and b = 3. This tells us the lengths of the semi-major and semi-minor axes. Therefore, the major axis runs vertically (along the y-axis) with a length of 2 and the minor axis runs horizontally (along the x-axis) with a length of 6.
04

Plot the axes

Plot the axes of the ellipse. Since the center is at the origin (0,0), plot the endpoints of the major axis at (0,1) and (0,-1) and the endpoints of the minor axis at (3,0) and (-3,0).
05

Draw the ellipse

Connect the plotted points smoothly to form the shape of the ellipse. Ensure that the ellipse touches these points and has the characteristic shape.

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Key Concepts

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

Ellipse Equation
An ellipse is a shape that looks like an elongated circle. In mathematics, its equation represents the set of all points where the sum of the distances from two fixed points, called foci, is constant.
In our exercise, the given equation is: \[ 9y^{2} = 9 - x^{2} \].
To graph this, we first need to rewrite it in a recognizable form. We start by dividing both sides by 9: \[ y^{2} = 1 - \frac{x^{2}}{9} \].
This can be rewritten into the standard form of an ellipse, which brings us to the next core concept.
Standard Form of Ellipse
The standard form equation of an ellipse is very helpful for graphing and identifying key properties. It鈥檚 given by: \[ \frac{y^{2}}{a^{2}} + \frac{x^{2}}{b^{2}} = 1 \].
This equation tells us how the ellipse is positioned and how it stretches along the x-axis and y-axis. In our case, after rewriting, it becomes: \[ \frac{y^{2}}{1} + \frac{x^{2}}{9} = 1 \].
In this equation, \( a^{2} = 1\) and \( b^{2} = 9\). So, \( a = 1 \) and \( b = 3 \).
This identification will be useful when we discuss the axes of the ellipse next.
Axes of Ellipse
The lengths and direction of the axes are crucial to drawing an accurate ellipse. For the equation \[ \frac{y^{2}}{a^{2}} + \frac{x^{2}}{b^{2}} = 1 \], the values of \( a \) and \( b \) tell us the lengths of the semi-major and semi-minor axes. In our example, \( a = 1 \) and \( b = 3 \), so:
  • The semi-major axis (along y-axis) runs from \( -1 \) to \( 1 \) giving us a total length of 2.
  • The semi-minor axis (along x-axis) stretches from \( -3 \) to \( 3 \) resulting in a total length of 6.
Plot these points on a coordinate plane:
  • Endpoints of the major axis at (0,1) and (0,-1).
  • Endpoints of the minor axis at (3,0) and (-3,0).
Finally, connect these points smoothly to form the ellipse. Remember, it will touch at all these points and follow the characteristic oval shape of an ellipse.

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