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Chapter 11: Problem 9

Given vertices \((\pm a, 0)\) and eccentricity \(e,\) what are the coordinates of the foci of an ellipse and a hyperbola?

Short Answer

Expert verified
Answer: The coordinates of the foci for both the ellipse and the hyperbola are \((\pm ea, 0)\).

Step by step solution

01

Find the distance between vertices and the center of the ellipse and hyperbola

Given the vertices \((\pm a, 0)\), we know that the distance between the two vertices is \(2a\). The center of the ellipse and hyperbola is the midpoint between the vertices, which is \((0, 0)\). This will be helpful when we calculate the distance between the center and the foci.
02

Recall the definition of eccentricity for an ellipse and a hyperbola

The eccentricity (\(e\)) of a conic section determines the shape of the curve. For an ellipse with semi-major axis, \(a\), and semi-minor axis, \(b\), the eccentricity can be defined as \(e = \sqrt{1 - \frac{b^2}{a^2}}\). On the other hand, for a hyperbola with the same semi-major axis, \(a\), and semi-minor axis, \(b\), the eccentricity can be defined as \(e = \sqrt{1 + \frac{b^2}{a^2}}\).
03

Calculate the distance between the center and the foci of an ellipse and a hyperbola

For an ellipse, the distance between the center and the foci, denoted as \(c\), can be given by \(c = ea\). And for a hyperbola, the distance between the center and the foci can also be given by \(c = ea\).
04

Find the coordinates of the foci of the ellipse and the hyperbola

For the ellipse, since the center is \((0, 0)\) and the distance between the center and the foci is \(c\), the coordinates of the foci are \((\pm c, 0)\) or \((\pm ea, 0)\). For the hyperbola, the center is also \((0, 0)\) and the distance between the center and the foci is \(c\), so the coordinates of the foci are \((\pm c, 0)\) or \((\pm ea, 0)\). The coordinates of the foci of the ellipse and the hyperbola are the same: \((\pm ea, 0)\).

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Most popular questions from this chapter

Consider the curve \(r=f(\theta)=\cos \left(a^{\theta}\right)-1.5\) where \(a=(1+12 \pi)^{1 / 2 \pi} \approx 1.78933\) (see figure). a. Show that \(f(0)=f(2 \pi)\) and find the point on the curve that corresponds to \(\theta=0\) and \(\theta=2 \pi\) b. Is the same curve produced over the intervals \([-\pi, \pi]\) and \([0,2 \pi] ?\) c. Let \(f(\theta)=\cos \left(a^{\theta}\right)-b,\) where \(a=(1+2 k \pi)^{1 / 2 \pi}, k\) is an integer, and \(b\) is a real number. Show that \(f(0)=f(2 \pi)\) and that the curve closes on itself. d. Plot the curve with various values of \(k\). How many fingers can you produce?

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