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Chapter 10: Problem 1

Conic Sections State the definitions of parabola, ellipse, and hyperbola in your own words.

Short Answer

Expert verified
A parabola is formed by the points equidistant from a focus and a directrix. An ellipse has all points such that the sum of distances to two foci is constant. A hyperbola has all points such that the difference of distances to two foci is constant.

Step by step solution

01

Define the Parabola

A parabola is the set of all points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). They are generated when a right circular cone is intersected with a plane parallel to the side of a cone.
02

Define the Ellipse

An ellipse is a set of points in a plane such that the sum of the distances from two fixed points (called the foci) is constant. This constant sum is the length of the major axis. Ellipses are formed when a right circular cone is intersected with a plane at an angle to its base but not parallel to its side.
03

Define the Hyperbola

A hyperbola is the set of all points in a plane such that the absolute difference of the distances to two fixed points (called the foci) is constant. Like ellipses, this constant difference is the length of the major axis but unlike ellipses, hyperbolas have two distinct disjoint curves. They are formed when a right circular cone is intersected with a plane at an angle to the base which is greater than the cone's angle itself.

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Key Concepts

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

Understanding Parabolas
A parabola is a fascinating shape seen in many areas like physics and engineering. It consists of all points that are equidistant from a fixed point known as the "focus" and a straight line called the "directrix." This means that each point on the parabola is the same distance from the focus as it is from the directrix.

Imagine slicing a cone parallel to its side; this slice reveals a parabolic shape. Parabolas have a unique U-shape, and can open upwards, downwards, or even sideways depending on the orientation.
  • They are symmetrical, with this symmetry centered on a line called the "axis of symmetry."
  • At the point where the parabola changes direction is the "vertex," the peak or lowest point.
  • Parabolas model real-world scenarios like the path of a thrown ball, reflecting dishes, and the design of car headlights.
Exploring Ellipses
Ellipses have a graceful, oval shape defined by two special points called "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 always constant. This provides a gentle, closed curve that wraps around the foci.

When you slice a cone at an angle that is not parallel to the base or the side, you create an ellipse.
  • The longest diameter of the ellipse is known as the "major axis," while the shortest is the "minor axis."
  • This unique shape is seen in planetary orbits, as many orbits are elliptical with the sun at one focus.
  • The symmetry of an ellipse comes from the perpendicular axes passing through its center.
Delving into Hyperbolas
Hyperbolas offer an intriguing structure formed by intersecting a cone with a plane steeper than its sides. They consist of two open, mirror-image curves. These curves are related by their distances from two fixed points, known as "foci." The unique property of hyperbolas is that the difference in distances from the foci to any point on the curve remains constant.

Unlike ellipses, a hyperbola has two separate branches that open away from each other.
  • The central line dividing the two branches is called the "transverse axis."
  • Hyperbolas are useful in various fields like physics, as they describe phenomena such as the paths of certain celestial objects and signals traveling through space.
  • In everyday applications, hyperbolas appear in the design of cooling towers and appear in specific lens designs for telescopes.

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Most popular questions from this chapter

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. Common interior of \(r=3-2 \sin \theta\) and \(r=-3+2 \sin \theta\)

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. Common interior of \(r=4 \sin 2 \theta\) and \(r=2\)

Finding a Polar Equation In Exercises \(33-38\) , find a polar equation for the conic with its focus at the pole and the given eccentricity and directrix. (For convenience, the equation for the directrix is given in rectangular form.) $$\begin{array}{ll}{\text { Conic }} & {\text { Eccentricity }} \\ {\text { Hyperbola}} & {\quad e=\frac{4}{3} }\end{array} \begin{array}{l}{\text { Directrix }} \\ {x=-2}\end{array}$$

Finding an Angle In Exercises \(107-112,\) use the result of Exercise 106 to find the angle \(\psi\) between the radial and tangent lines to the graph for the indicated value of \(\theta\) . Use a graphing utility to graph the polar equation, the radial line, and the tangent line for the indicated value of \(\theta .\) Identify the angle \(\psi\) . $$r=3(1-\cos \theta) \quad \theta=\frac{3 \pi}{4}$$

Comet Hale-Bopp The comet Hale-Bopp has an elliptical orbit with the sun at one focus and has an eccentricity of \(e \approx 0.995 .\) The length of the major axis of the orbit is approximately 500 astronomical units. (a) Find the length of its minor axis. (b) Find a polar equation for the orbit. (c) Find the perihelion and aphelion distances.
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