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Chapter 9: Problem 12

Graph each ellipse and locate the foci. $$y^{2}=1-4 x^{2}$$

Short Answer

Expert verified
The graph of the equation is a degenerate ellipse or a line segment along the y-axis between points (0,1) and (0,-1). The foci are not located on the real plane due to the resulting imaginary number from the calculation in step 2.

Step by step solution

01

Identify the Major and Minor Axes

First, rewrite the given equation in the standard form of an ellipse, which is \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). This can be done by dividing every term in the original equation by 1. This gives us \(4x^{2}+y^{2}=1\). The denominator under the \(x\) term is the square of the length of the semi-minor axis (which we will denote as \(b\)), and the denominator under the \(y\) term is the square of the length of the semi-major axis (which we will denote as \(a\)). So, we can conclude \(b = \sqrt{4} = 2\) (semi-minor axis) and \(a = \sqrt{1} = 1\) (semi-major axis).
02

Find the Foci

To find the foci of the ellipse, we use the relationship given by \(c=\sqrt{a^{2}-b^{2}}\), where \(c\) is the distance from the center of the ellipse to either focus. Substituting the values of \(a\) and \(b\) found in Step 1, we get \(c=\sqrt{1^{2}-2^{2}} = \sqrt{-3}\). Since \(c\) is imaginary, it signifies that the foci of the ellipse fall on the imaginary axis. This shows the ellipse degenerates into a line segment along the y-axis.
03

Plot the Ellipse and the Foci

We start by marking out the major and minor axes on the Cartesian plane. Along the y-axis (the major axis), we mark the points (0,1) and (0,-1). Along the x-axis (the minor axis), we mark the points (-2,0) and (2,0). We also need to mention that \(c\) is an imaginary number, so it doesn't contribute to the real part of the graph, creating the degenerate line segment.

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Key Concepts

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

Foci of an Ellipse
In the study of ellipses, the foci are two special points located along the major axis. These points hold significant importance as the sum of the distances from any point on the ellipse to the foci is constant. In this exercise, we found the formula for the distance to the foci using the equation \( c = \sqrt{a^2 - b^2} \). However, it's important to note that in some cases, like this one, the value of \( c \) can be imaginary. This happens when the semi-major axis \( a \) is smaller than the semi-minor axis \( b \). In our scenario, with \( c = \sqrt{1^2 - 2^2} = \sqrt{-3} \), the foci don't exist in the real plane, indicating that the ellipse degenerates into a different geometric shape. Although imaginary foci m

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