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Figure 13-7
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A basic hyperbola in the Cartesian xy plane The eccentricity is greater than 1 The distances a and b are the semi-axes
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You might ask, Is it possible to have a conic section with negative eccentricity For our purposes in this course, the answer is no Negative eccentricity involves the notion of negative distances If we allow the eccentricity of a noncircular conic section to become negative, we get an inside-out ellipse, parabola, or hyperbola In ordinary geometry, such a curve is the same as a real-world ellipse, parabola, or hyperbola That said, it s worth noting that in certain high-level engineering and physics applications, negative distances sometimes behave differently than positive distances In those special situations, an inside-out conic section might represent an entirely different phenomenon from a real-world conic section Keep that in the back of your mind if you plan on becoming an astronomer, cosmologist, or high-energy physicist someday!
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Using the alternative formula for the eccentricity of an ellipse, show that if we have an ellipse in which e = 0, then that ellipse is a circle
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Solution
First, we should realize that a circle is a special sort of ellipse in which the two semi-axes have identical length With that in mind, let s plug e = 0 into the alternative equation for the eccentricity of an ellipse That gives us 0 = d /(2s) where d is the distance between the foci, and s is the length of the major semi-axis We can multiply the above formula by 2s to obtain 0=d which tells us that the two foci are located at the same point, so there s really only one focus A circle is the only type of ellipse that has a single focus
Standard Equations
When we graph a conic section in the Cartesian xy plane, we can find a unique equation that represents that curve These equations are always of the second degree, meaning that the equation must contain the square of one or both variables
Equations for circles We can write the standard-form general equation for a circle in terms of its center point and its radius as
(x x0)2 + ( y y0)2 = r2 where x0 and y0 are real constants that tell us the coordinates (x0,y0) of the center of the circle, and r is a positive real constant that tells us the radius (Fig 13-8) When a circle is centered at the origin, the equation is simpler because x0 = 0 and y0 = 0 Then we have x2 + y2 = r2 The simplest possible case is the unit circle, centered at the origin and having a radius equal to 1 Its equation is x 2 + y2 = 1
Equations for ellipses The standard-form general equation of an ellipse in the Cartesian xy plane, as shown in Fig 13-9, is
(x x0)2/a2 + ( y y0)2/b2 = 1
Standard Equations
(x0, y0) r
Figure 13-8 Graph of the circle for (x x0)2 + (y y0)2 = r2
(x0, y0)
Figure 13-9
Graph of the ellipse for (x x0)2/a2 + ( y y0)2/b2 = 1
Conic Sections
where x0 and y0 are real constants representing the coordinates (x0,y0) of the center of the ellipse, a is a positive real constant that represents the distance from (x0,y0) to the curve along a line parallel to the x axis, and b is a positive real constant that tells us the distance from (x0,y0) to the curve along a line parallel to the y axis When we plot x on the horizontal axis and y on the vertical axis (the usual scheme), a is the length of the horizontal semi-axis or horizontal radius of the ellipse, and b is the length of the vertical semi-axis or vertical radius For ellipses centered at the origin, we have x0 = 0 and y0 = 0, so the general equation is x2/a2 + y2/b2 = 1 If a = b, then the ellipse is a circle Remember that a circle is an ellipse for which the eccentricity is 0
Equations for parabolas Figure 13-10 is an example of a parabola in the Cartesian xy plane The standard-form general equation for this curve is
y = ax2 + bx + c The vertex is at the point (x0,y0) We can find these values according to the formulas x0 = b /(2a)
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