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Chapter 4: Problem 46

Identify and sketch the graph of the conic section. $$ 5 x^{2}+3 y^{2}-15=0 $$

Short Answer

Expert verified
The graph of the conic section is an ellipse with center at (0,0), a horizontal axis length of \(\sqrt{3}\), and a vertical axis length of \(\sqrt{5}\).

Step by step solution

01

Identify the type of conic section

In the equation \(5x^{2}+3y^{2}-15=0\), both x and y are squared and these squared terms have the same sign, therefore the conic section is an ellipse.
02

Rewrite the equation in standard form

The standard form of an ellipse is \((x-h)^{2}/a^{2} + (y-k)^{2}/b^{2} = 1\), where (h,k) is the center and a and b are the lengths of the semi-major and semi-minor axes. First, rearrange the given equation so that it equals 1: \(5x^{2}/15 + 3y^{2}/15 = 1\), which simplifies to \(x^2/3 + y^2/5 = 1\). So h = 0, k = 0, a = \sqrt{3}, and b = \sqrt{5}.
03

Sketch the graph based on the standard form

Plot the center of the ellipse at the origin (0,0). The major axis is vertical because b > a, and it extends \sqrt{5} units above and below the center. The minor axis is horizontal and extends \sqrt{3} units to the left and right of the center. Sketch the ellipse using these axes as guides.

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Key Concepts

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

Ellipse
An ellipse is a type of conic section, which is a curve obtained by slicing a right circular cone with a plane. An ellipse appears as an elongated circle and can be described as the set of all points for which the sum of the distances to two fixed points, called the foci, is constant. This gives ellipses their distinctive, stretched look, compared to perfect circles which are a simple special case of an ellipse.

Some essential properties of ellipses include:
  • Axes: They have two axes, the major and minor axes, with the major axis being the longest diameter.
  • Foci: Each ellipse has two focal points. The closer these points are to the center, the more the ellipse resembles a circle.
  • Center: The midpoint of the line segment joining the foci, often used as a reference point for graphing.
Understanding these properties can help you identify and graph ellipses with greater ease.
Standard Form of Ellipse
The standard form of an ellipse is a mathematical representation that helps in understanding and graphing this shape. It allows us to easily derive information about the ellipse, such as its center and the lengths of its axes.

The equation of an ellipse in standard form is\[(x-h)^2/a^2 + (y-k)^2/b^2 = 1\]

Where:
  • (h, k): The coordinates of the center of the ellipse.
  • a: The semi-major axis, representing half the longest diameter of the ellipse.
  • b: The semi-minor axis, representing half the shortest diameter of the ellipse.
To convert a quadratic equation into this form, you typically need to divide all terms to ensure the equation equals 1, as seen in the rewritten equation for our ellipse \(x^2/3 + y^2/5 = 1\). This conversion simplifies the process of determining important characteristics of the ellipse.
Graphing Ellipses
Graphing ellipses involves understanding their geometry and plotting them correctly on a coordinate plane. It's crucial to start with the standard form of the ellipse equation as it provides direct insights into the ellipse's dimensions and position.

Here are steps to graph an ellipse:
  • Locate the Center: Identify the center (h, k) from the standard form equation. This point serves as the starting point for your graph.
  • Draw the Axes: Determine the lengths of the semi-major and semi-minor axes from the values of a and b. The major axis corresponds to the greater value between a or b. The ellipse's orientation depends on whether \(a > b\) (horizontal) or \(b > a\) (vertical).
  • Plot the Axes: From the center, draw lines parallel to the x and y axes according to the values of a and b. These lines will guide the shape of your ellipse.
  • Sketch the Ellipse: Use the plotted axes to draw the smooth, oval shape of the ellipse, ensuring the curve smoothly extends to the endpoints of the major and minor axes.
By following these steps, you can accurately graph ellipses, visualizing their unique shape and properties more effectively on paper or digitally.

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Most popular questions from this chapter

Identify and sketch the graph of the conic section. $$ y^{2}+x=0 $$

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