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Chapter 10: Problem 78

Conjecture (a) Show that the equation of an ellipse can be written as \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{a^{2}\left(1-e^{2}\right)}=1\) (b) Use a graphing utility to graph the ellipse \(\frac{(x-2)^{2}}{4}+\frac{(y-3)^{2}}{4\left(1 e^{2}\right)}=1\) for \(e=0.95, e=0.75, e=0.5, e=0.25,\) and \(e=0\) (c) Use the results of part (b) to make a conjecture about the change in the shape of the ellipse as \(e\) approaches \(0 .\)

Short Answer

Expert verified
a) Verified the given equation is a form of the general equation of an ellipse. b) Upon graphing we realize as 'e' decreases the ellipse becomes more like a circle. c) Conjecture - 'As e approaches 0, the ellipse will become increasingly circular.'

Step by step solution

01

Verifying the equation of an ellipse

The general equation of an ellipse (centered at (h,k) instead of at the origin) is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where 'a' and 'b' are lengths of the semi-major and semi-minor axes, respectively. This is clear that the provided equation is a form of this, where \(b^2 = a^2(1 - e^2)\) instead. So part (a) is confirmed.
02

Graphing the ellipse for different eccentricities

Using a graphing utility, graph the given equation for the different values of 'e' as assigned in the exercise. You will observe the change in the shape of the ellipse for different values of e
03

Making a conjecture from observation

From observing the graphs made in step 2, as 'e' approaches 0, the ellipse appears increasingly like a circle. This suggests that the value of 'e' relates to how 'elliptical' the shape is - the closer 'e' is to 0, the more circular it is; the closer to 1, the more stretched out or 'elliptical' it is. So for part (c), a valid conjecture could be 'As e approaches 0, the ellipse will become increasingly circular.'

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Key Concepts

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

Graphing Ellipses
Graphing an ellipse is a visual way to understand its geometric shape and properties. To graph an ellipse centered around the point \((h, k)\), you need to identify the lengths of its semi-major axis (a) and semi-minor axis (b). The general equation for an ellipse centered at (h, k) is given as \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\).

When graphing, you plot the center at (h, k), then mark points a units to the left and right of the center for the ends of the horizontal major axis, and b units up and down for the ends of the vertical minor axis. If \(a > b\), the ellipse stretches further horizontally, and if \(a < b\), it stretches more vertically. After plotting these points, you draw a smooth, oval shape connecting these points, making sure it encompasses the center of the ellipse.
Eccentricity of an Ellipse
The eccentricity of an ellipse, typically denoted as \(e\), is a measure of how much the shape of an ellipse deviates from being a circle. It takes on values between 0 and 1 where an \(e\) of 0 corresponds to a perfect circle, as the eccentricity approaches 1, the ellipse becomes more elongated.

In the equation \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\), if you replace \(b^{2}\) with \(a^{2}(1 - e^{2})\), you get an equation that directly incorporates the eccentricity.

Eccentricity and Graph Shapes

As seen in the textbook exercise, when you graph several ellipses with the same centers and major axes but different \(e\) values, the ones with lower eccentricities will appear more circular, while those with higher eccentricities will appear more stretched along the major axis. Understanding this concept allows you to predict the shape of an ellipse simply by knowing its eccentricity.
Conic Sections
Conic sections are the curves obtained by intersecting a plane with a double-napped cone. They are classified based on the angle of the plane relative to the cone's surface, which can result in different shapes: circles, ellipses, parabolas, and hyperbolas.

An ellipse is one type of conic section, resulting from a plane cutting through the cone at an angle that is less than that made by the side of the cone, but not parallel to its base.

The Role of Eccentricity

Eccentricity plays a crucial role in conic sections, determining their specific shapes. For instance, a conic section with an eccentricity of 0 is a circle, while an eccentricity between 0 and 1 corresponds to an ellipse. The insight gained from the elliptical eccentricity helps to classify and understand the properties of these conic shapes in a broader mathematical and geometric context.

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Most popular questions from this chapter

Integration by Substitution Use integration by substitution to show that if \(y\) is a continuous function of \(x\) on the interval \(a \leq x \leq b,\) where \(x=f(t)\) and \(y=g(t),\) then $$\begin{array}{l}{\int_{a}^{b} y d x=\int_{t_{1}}^{t_{2}} g(t) f^{\prime}(t) d t} \\ {\text { where } f\left(t_{1}\right)=a, f\left(t_{2}\right)=b, \text { and both } g \text { and } f^{\prime} \text { are continuous }} \\ {\text { on }\left[t_{1}, t_{2}\right].}\end{array}$$

Antenna Radiation The radiation from a transmitting antenna is not uniform in all directions. The intensity from a particular antenna is modeled by\(r=a \cos ^{2} \theta\) (a) Convert the polar equation to rectangular form. (b) Use a graphing utility to graph the model for \(a=4\) and \(a=6 .\) (c) Find the area of the geographical region between the two curves in part (b).

Finding the Area of a Polar Region Between Two Curves In Exercises \(37-44\) , use a graphing utility to graph the polar equations. Find the area of the given region analytically. Inside \(r=3 \sin \theta\) and outside \(r=1+\sin \theta\)

Eccentricity In Exercises 67 and 68 , let \(r_{0}\) represent the distance from a focus to the nearest vertex, and let \(r_{1}\) represent the distance from the focus to the farthest vertex. Show that the eccentricity of an ellipse can be written as \(e=\frac{r_{1}-r_{0}}{r_{1}+r_{0}}\) Then show that \(\frac{r_{1}}{r_{0}}=\frac{1+e}{1-e}\)

True or False? In Exercises \(113-116,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) represent the same point on the polar coordinate system, then \(\left|r_{1}\right|=\left|r_{2}\right|\)
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