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Chapter 13: Problem 17

Graph each of the following equations. $$ 2 x^{2}+3 y^{2}=6 $$

Short Answer

Expert verified
Rewrite the equation as \( \frac{x^2}{3} + \frac{y^2}{2} = 1 \) and plot with axes lengths of 1.73 and 1.41.

Step by step solution

01

Identify the type of conic section

To determine the type of conic section represented by the given equation, locate the coefficients of the squared terms. The equation is given as: \[ 2x^2 + 3y^2 = 6 \]Since both terms involving variables are squared and have positive coefficients but different values, the equation represents an ellipse.
02

Rewrite the equation in standard form

Divide both sides of the equation by 6 to rewrite it in standard form. This helps to identify the lengths of the semi-major and semi-minor axes:\[ \frac{2x^2}{6} + \frac{3y^2}{6} = \frac{6}{6} \]Simplify each fraction:\[ \frac{x^2}{3} + \frac{y^2}{2} = 1 \]
03

Identify the lengths of the axes

From the standard form, \( \frac{x^2}{3} + \frac{y^2}{2} = 1 \), we can identify the lengths of the semi-major and semi-minor axes. The equation can be written as:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where \(a^2 = 3\) and \(b^2 = 2\). Therefore, \(a = \sqrt{3} \approx 1.73\) and \(b = \sqrt{2} \approx 1.41\).
04

Plot the ellipse using the calculated axes

To graph the ellipse, plot the center at the origin (0,0). The semi-major axis (\(a\)) extends approximately 1.73 units along the x-axis in both positive and negative directions. The semi-minor axis (\(b\)) extends approximately 1.41 units along the y-axis in both positive and negative directions. Draw the ellipse by connecting these points in a smooth, oval shape.

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Key Concepts

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

Conic Sections
Conic sections are curves formed by the intersection of a plane and a double-napped cone. These curves include circles, ellipses, parabolas, and hyperbolas. To classify a given equation as a type of conic section, examine the coefficients of the squared terms. If both squared terms in an equation have positive coefficients and different values, the curve is an ellipse.
Standard Form of Ellipse
The standard form of the equation for an ellipse is:
\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
This equation makes it clear how the axes are scaled. Compare it to your given equation to rewrite it in standard form. Divide by the constant term, simplify, and identify values representing the axes' lengths. For instance, \( \frac{2x^2}{6} + \frac{3y^2}{6} = 1 \) can be simplified to \( \frac{x^2}{3} + \frac{y^2}{2} = 1 \).
Semi-Major Axis
In an ellipse, the longest radius is the semi-major axis. It extends from the center to the ellipse's perimeter along the ellipse's longest diameter. The length of the semi-major axis is denoted by \(a\). In the standard form equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the value of \ a \ is derived by taking the square root of the denominator under x's term. For the equation \( \frac{x^2}{3} + \frac{y^2}{2} = 1 \), \( \sqrt{3} \approx 1.73\).
Semi-Minor Axis
While the semi-major axis is the ellipse's longest radius, the semi-minor axis is the shortest. It runs from the center to the ellipse's edge along the shortest diameter. In the standard form equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the value of \(b\) is determined by the square root of the denominator under y's term. For \( \frac{x^2}{3} + \frac{y^2}{2} = 1 \), the semi-minor axis length is \( \sqrt{2} \approx 1.41\).

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