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Chapter 11: Problem 1

Give the property that defines all parabolas.

Short Answer

Expert verified
#Answer# The focus-directrix property defines all parabolas, regardless of their orientation or dimensions. It states that each point on the parabola is equidistant from a fixed point (the focus) and a fixed line (the directrix).

Step by step solution

01

Definition of a Parabola

A parabola is a set of points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). This is known as the focus-directrix property.
02

Identify the Focus and Directrix

Firstly, mention the focus, denoted as F, which is a fixed point in the plane. Then, mention the directrix, which is a fixed line in the plane that is not on the parabola itself.
03

Describe the Relationship between a Point on the Parabola, the Focus, and the Directrix

Now, let's pick any point P on the parabola. The relationship we need to establish is that the distance from point P to the focus F is equal to the distance from point P to the directrix. This can be written as PF = PD, where D is the point on the directrix which is perpendicular to point P.
04

Explore Different Orientations of Parabolas

The focus-directrix property holds true for all types of parabolas, regardless of their orientation. A parabola can either be vertical or horizontal, and its focus and directrix will be positioned accordingly. However, the focus-directrix property remains consistent in all cases, which is that every point on the parabola equidistant to the focus and the directrix.
05

Use the Property to Define Parabolas

To find the equation of a parabola given its focus and directrix, use the distance formula and equate the distance from the focus, F(h,k), to a point (x,y) on the parabola with the distance from the same point (x,y) to the directrix. Solve this equation to obtain either a standard form or vertex form equation for the parabola.
06

Conclusion

In conclusion, the property that defines all parabolas is the focus-directrix property, which states that each point on the parabola is equidistant from its focus and directrix. This property holds true for all types of parabolas, regardless of their orientation or dimensions.

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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 lines tangent to the endpoints of any focal chord of a parabola \(y^{2}=4 p x\) intersect on the directrix and are perpendicular.

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 \(n\) for determining the least positive number \(P\) such that the complete curve is generated over the interval \([0, P]\)

Sketch the graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. $$8 y=-3 x^{2}$$

Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola that opens downward with directrix \(y=6\)

Sketch the graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. $$4 x^{2}-y^{2}=16$$
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