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Chapter 7: Problem 61

What is an ellipse?

Short Answer

Expert verified
An ellipse is a curve that is the locus of all points in the plane the sum of whose distances from two fixed points (the foci) is a constant. It is characterized by two axes (major and minor) and two foci.

Step by step solution

01

Define an Ellipse

An ellipse is a type of conic section that is not a circle but closely related to it. This shape is formed by a plane intersecting a cone. It is a curve consisting of all points for which the sum of its distances from two fixed points, called foci, is constant.
02

Describe the Properties of an Ellipse

The main properties of an ellipse include: 1) The sum of distances of any point on the ellipse from two fixed points (foci) is always constant. 2) The longest diameter is called the major axis, while the shortest diameter is the minor axis. 3) The points at the ends of the major axis are called vertices.
03

Explanation of Major and Minor Axes

The Major Axis is the longest diameter of the ellipse. It passes through the foci. The minor Axis is the shortest diameter of the ellipse. It is perpendicular to the major axis.
04

Explanation of Foci

The two fixed points inside the ellipse are called foci. They are always located along the major axis. For any point on the ellipse, the sum of the distances to the foci is the same.

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Most popular questions from this chapter

Which one of the following is true? a. The parabola whose equation is \(x=2 y-y^{2}+5\) opens to the right. b. If the parabola whose equation is \(x=a y^{2}+b y+c\) has its vertex at \((3,2)\) and \(a>0,\) then it has no \(y\) -intercepts. c. Some parabolas that open to the right have equations that define \(y\) as a function of \(x .\) d. The graph of \(x=a(y-k)+h\) is a parabola with vertex at \((h, k)\)

What is a hyperbola?

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$(x-3)^{2}-4(y+3)^{2}=4$$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$\frac{x^{2}}{\frac{81}{4}}+\frac{y^{2}}{\frac{25}{16}}=1$$

Graph each ellipse and give the location of its foci. $$36(x+4)^{2}+(y+3)^{2}=36$$
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