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Chapter 13: Problem 93

What is a parabola?

Short Answer

Expert verified
A parabola is a U-shaped curve that is mirror-symmetrical. It is the locus of all points in the plane equidistant from a fixed point (the focus) and a fixed line (the directrix). The parabola can open upwards, downwards, or sideways. Parabolas have practical applications in multiple fields such as physics, engineering, and astronomy.

Step by step solution

01

Definition of a Parabola

A parabola is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented either up or down, or L-shaped when oriented either side. It fits any of several superficially different mathematical descriptions which can all be proved to define exactly the same curves.
02

Mathematical Description

In mathematics terms, a parabola is the set of all points in the plane equidistant from a given fixed point (the focus) and given fixed line (the directrix). Therefore, a parabola can be defined mathematically as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.
03

Graphical Representation

Visually, a parabola looks like a U-shaped curve. The vertex of the parabola is the point that splits the parabola into two symmetric halves, and the axis of symmetry goes through the vertex and the focus pointing upwards if the parabola opens upwards (or downwards if the parabola opens downwards). The directrix is a line perpendicular to the axis of symmetry, located such that it is equidistant from the vertex as the focus, but lies on the opposite side of the parabola.
04

Usage of Parabolas

Parabolas are used all the time in daily life and have practical applications in fields like physics, engineering, astronomy, architecture, and sports. For example, the path of any projectile in motion (disregarding wind resistance and other forces) is a parabola, and the reflective surfaces of mirrors used in telescopes or headlights are often parabolic.

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Most popular questions from this chapter

(GRAPH CANT COPY) Find the coordinates of the vertex for the horizontal parabola defined by the given equation. $$x=-2(y+5)^{2}-1$$

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$\left\\{\begin{array}{l}x=y^{2}-5 \\ x^{2}+y^{2}=25\end{array}\right.$$

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$\left\\{\begin{array}{l}x=2 y^{2}+4 y+5 \\\ (x+1)^{2}+(y-2)^{2}=1\end{array}\right.$$

Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola. Then graph the conic section. $$4 x^{2}+y^{2}=16$$

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$\left\\{\begin{array}{l}x=(y-3)^{2}+2 \\ x+y=5\end{array}\right.$$
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