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Chapter 10: Problem 12

How does the eccentricity determine the type of conic section?

Short Answer

Expert verified
Answer: Parabola

Step by step solution

01

Define eccentricity

Eccentricity is a dimensionless quantity that describes the degree to which a conic section deviates from a circle. Mathematically represented by \(e\), eccentricity can have a value in the range of \(0 \leq e \leq \infty\).
02

Analyze the relationship between eccentricity and conic sections

Depending on the value of the eccentricity, we can determine the type of conic section as follows: 1. e = 0: A circle 2. 0 < e < 1: An ellipse (excluding circles) 3. e = 1: A parabola 4. e > 1: A hyperbola Now let's discuss each case in detail:
03

Eccentricity and circles

If the eccentricity \(e\) is equal to 0, it means that the conic section has no deviation from a circle. In this case, we have the equation of a circle, which can be written as: \[(x-a)^2 +(y-b)^2=r^2\] where \((a, b)\) is the center of the circle, and \(r\) is its radius.
04

Eccentricity and ellipses

When the eccentricity \(e\) lies between 0 and 1 (excluding 0), we have an ellipse. The general equation of an ellipse is given by: \[\frac{(x-a)^2}{h^2}+\frac{(y-b)^2}{k^2}=1\] where \((a, b)\) is the center of the ellipse, \(h\) is the distance from the center to the vertex on the major axis, and \(k\) is the distance from the center to the vertex on the minor axis.
05

Eccentricity and parabolas

If the eccentricity \(e\) equals 1, then the conic section is a parabola. The equation of a parabola can be written in either of the following forms: \[y=ax^2+bx+c\] or \[x=ay^2+by+c\] where \(a\), \(b\), and \(c\) are constants.
06

Eccentricity and hyperbolas

When the eccentricity \(e\) is greater than 1, the conic section is a hyperbola. The general equation of a hyperbola is given by: \[\frac{(x-a)^2}{h^2}-\frac{(y-b)^2}{k^2}=1\] or \[\frac{(y-b)^2}{k^2}-\frac{(x-a)^2}{h^2}=1\] where \((a, b)\) is the center of the hyperbola, \(h\) is the distance from the center to a vertex on the transverse axis, and \(k\) is the distance from the center to a vertex on the conjugate axis.

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Most popular questions from this chapter

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The length of the latus rectum of an ellipse centered at the origin is \(2 b^{2} / a=2 b \sqrt{1-e^{2}}\)

An ellipse (discussed in detail in Section 10.4 ) is generated by the parametric equations \(x=a \cos t, y=b \sin t.\) If \(0 < a < b,\) then the long axis (or major axis) lies on the \(y\) -axis and the short axis (or minor axis) lies on the \(x\) -axis. If \(0 < b < a,\) the axes are reversed. The lengths of the axes in the \(x\) - and \(y\) -directions are \(2 a\) and \(2 b,\) respectively. Sketch the graph of the following ellipses. Specify an interval in t over which the entire curve is generated. $$x=4 \cos t, y=9 \sin t$$

Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y.\) $$x=\sqrt{t+1}, y=\frac{1}{t+1}$$

Find real numbers a and b such that equations \(A\) and \(B\) describe the same curve. \(A: x=10 \sin t, y=10 \cos t ; 0 \leq t \leq 2 \pi\) \(B: x=10 \sin 3 t, y=10 \cos 3 t ; a \leq t \leq b\)

Consider the polar curve \(r=\cos (n \theta / m)\) where \(n\) and \(m\) are integers. a. Graph the complete curve when \(n=2\) and \(m=3\) b. Graph the complete curve when \(n=3\) and \(m=7\) c. Find a general rule in terms of \(m\) and \(n\) (where \(m\) and \(n\) have no common factors) for determining the least positive number \(P\) such that the complete curve is generated over the interval \([0, P]\).
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