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Chapter 10: Problem 4

Match the conic with its eccentricity. (a) \(01\) (i) parabola (ii) hyperbola (iii) ellipse

Short Answer

Expert verified
The matching pairs are: (a) corresponds to (iii), (b) corresponds to (i), (c) corresponds to (ii).

Step by step solution

01

Remember the eccentricity for each conic

For each type of conic section, recall the properties that define them: \n1. An ellipse is defined as the set of all points in the plane for which the sum of the distances to two fixed points (the foci) is a constant. It's eccentricity is given by \(0 < e < 1\). \n2. A parabola is defined as the set of all points in the plane that are equidistant from a fixed line (the directrix) and a fixed point (the focus). It's eccentricity is given by \(e = 1\). \n3. A hyperbola is defined as the set of all points in the plane for which the absolute difference of the distances to two fixed points is a constant. It's eccentricity is given by \(e > 1\).
02

Match the conic with its eccentricity

Now, we match each conic to its eccentricity using the properties we remembered from step 1: \n1. For \(01\), the conic is a hyperbola, so (c) corresponds to (ii).

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Key Concepts

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

Eccentricity
Eccentricity is a fundamental concept in understanding conic sections. It is a measure that distinguishes the shape of different conic sections.
  • An eccentricity of 0 corresponds to a circle, a special case of an ellipse.
  • Ellipse has an eccentricity between 0 and 1, representing that its shape is more like an elongated circle.
  • Parabola has an eccentricity of exactly 1, which gives it its unique open shape.
  • Hyperbola has an eccentricity greater than 1, indicating its two separate curves.
Eccentricity helps define each conic section, based on the ratio of the distances from a point on the conic to a focus, and from that point to a directrix. The way this ratio behaves or changes helps us categorize the form of the conic.
Ellipse
An ellipse is a conic section that appears like a stretched circle. It is defined by two fixed points known as foci. The defining property of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant. To visualize:
  • Picture a loop or track.
  • There are two points inside that track (the foci).
  • Every point on the edge of the loop is such that the sum of the distances to the foci remains the same.
The eccentricity of an ellipse, which ranges from 0 to 1, determines how "stretched" it looks. As the eccentricity approaches 0, the ellipse becomes more circular. As it approaches 1, it gets more elongated.
Parabola
A parabola is the set of all points that are equidistant from a fixed point, called the focus, and a fixed line, known as the directrix. This creates a mirror-like symmetrical curve with an eccentricity of exactly 1. In simple terms:
  • Imagine a U-shaped curve.
  • Positioned such that each point of this curve is the same distance from a single point (focus) and a line (directrix).
Parabolas are notable in physics and engineering, as they describe the path of objects under uniform gravity and no other forces, like a thrown ball. They show up in various real-world applications such as satellite dishes and car headlights that use their reflective properties.
Hyperbola
A hyperbola is an open curve formed by a plane that cuts both nappes of a double cone. Unlike ellipses and parabolas, hyperbolas have two separate branches. The key property:
  • A hyperbola is characterized by the difference of the distances from any point on the curve to two fixed points (foci) being constant.
With an eccentricity greater than 1, hyperbolas have a wide and open structure. Hyperbolas can be visualized as two mirrored curves or loops. These shapes prove significant in fields like astronomy and navigation, such as in the paths of celestial bodies or navigational systems. The properties of hyperbolas make them suitable for describing systems that exhibit diverging paths.

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Most popular questions from this chapter

Convert the polar equation to rectangular form. Then sketch its graph. $$r=3 \sec \theta$$

(a) Show that the distance between the points \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) is \(\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{1}-\theta_{2}\right)}\) (b) Simplify the Distance Formula for \(\theta_{1}=\theta_{2} .\) Is the simplification what you expected? Explain. (c) Simplify the Distance Formula for \(\theta_{1}-\theta_{2}=90^{\circ}\) Is the simplification what you expected? Explain.

Explain how the graph of each conic differs from the graph of \(r=\frac{5}{1+\sin \theta}\) (See Exercise 17.) (a) \(r=\frac{5}{1-\cos \theta}\) (b) \(r=\frac{5}{1-\sin \theta}\) (c) \(r=\frac{5}{1+\cos \theta}\) (d) \(r=\frac{5}{1-\sin [\theta-(\pi / 4)]}\)

Convert the polar equation to rectangular form. $$r=2 \sin 3 \theta$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc}\text{Conic} & \text{Vertex or Vertices} \\ \text{Ellipse} & (2,0),(10, \pi) \end{array}$$
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