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Chapter 4: Problem 45

Identify and sketch the graph of the conic section. $$ x^{2}+4 y^{2}-16=0 $$

Short Answer

Expert verified
The given equation represents an ellipse with semi-major axis of length 4, semi-minor axis of length 2 centered at the origin. Sketch the ellipse using these measures.

Step by step solution

01

Identify the type of conic section

To identify the type of conic section, look at the equation \( x^{2}+4 y^{2}-16=0 \). This equation is in the form of \(Ax^2 + By^2 = C\), where A and B are positive. This is the standard form for an ellipse.
02

Find the semi-axes lengths

Rewrite the equation in the standard form of an ellipse: \(\frac{x^{2}}{a^2}+\frac{y^{2}}{b^2}=1\). Comparing this with the given equation, it comes out as \(\frac{x^{2}}{16}+\frac{y^{2}}{4}=1\). So, \(a^2=16\), \(b^2=4\). Thus, the length of semi-major axis \(a\) is 4 and length of semi-minor axis \(b\) is 2.
03

Sketch the ellipse

Plot the center of the ellipse at the origin (0,0). From there, move 4 units to left and right to plot the ends of the major axis, and 2 units up and down to plot the ends of the minor axis. Finish it by sketching a smooth curve to connect these points and complete the ellipse.

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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 fascinating curves that can be formed by slicing through a cone. Each type of curve is unique and has its own set of properties. Understanding conic sections is essential in the study of geometry and its applications.
  • Circle: A special type of ellipse where the two axes are the same.
  • Ellipse: An elongated circle with two different axes, one longer than the other.
  • Parabola: A curve that mirrors on one side when dissected by a plane parallel to its side.
  • Hyperbola: Two disconnected curves formed by slicing a cone with a plane that has a steeper angle than the side of the cone.
Identifying which conic section is represented by an equation depends on its structure. In the form \(Ax^2 + By^2 = C\), if both \(A\) and \(B\) are positive, like in the given exercise, an ellipse is indicated. This is one of the primary properties that differentiate an ellipse from other conic sections.
Equation of an Ellipse
The equation of an ellipse defines its shape and orientation within a coordinate system. The standard form of an ellipse is given by \[ \frac{x^{2}}{a^2} + \frac{y^{2}}{b^2} = 1 \] where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. This equation is critical to visualize the curve and understand its symmetry around the central axis.
  • Horizontal ellipse: If \(a > b\), the ellipse stretches more along the x-axis.
  • Vertical ellipse: If \(b > a\), the ellipse is taller, stretching more along the y-axis.
By modifying the values of \(a\) and \(b\), you can manipulate the size and orientation of the ellipse. In the exercise, the equation \(\frac{x^{2}}{16} + \frac{y^{2}}{4} = 1\) indicates a horizontal ellipse, as \(a = 4\) is greater than \(b = 2\).
Semi-Major and Semi-Minor Axes
The axes of an ellipse play a crucial role in defining its dimensions. These axes are determined from the equation of the ellipse and assist in sketching the shape accurately. Below are important points regarding each type of axis:
  • Semi-Major Axis: The longest radius of the ellipse. It stretches from the center to the furthest point on the perimeter. In the equation \(a > b\), the semi-major axis will be parallel to the x-axis in a horizontal ellipse.
  • Semi-Minor Axis: The shortest radius of the ellipse that runs perpendicular to the semi-major axis. It stretches from the center to the shortest perimeter point.
Understanding these axes is crucial when sketching an ellipse, as they determine the distances you mark from the center to define the overall size and orientation. In the equation \(\frac{x^{2}}{16} + \frac{y^{2}}{4} = 1\), the semi-major axis has a length of 4 and the semi-minor axis has a length of 2, helping to accurately plot the curve.

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