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Chapter 7: Problem 57

What is a parabola?

Short Answer

Expert verified
A parabola is a U-shaped symmetrical curve, also known as a conic section, formed by the intersection of a cone and a plane. It's represented by the equation \( y = ax^2 + bx + c \) in Cartesian coordinates. A distinguishing property of a parabola is that any point on the parabola is equidistant from the focus and the directrix.

Step by step solution

01

Definition

A parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It is a specific type of curve called a conic section, formed by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.
02

Mathematical Description

In the Cartesian coordinate system, the graph of a quadratic equation \( y = ax^2 + bx + c \) is a parabola. The property that distinguishes a parabola is that, for any point on the parabola, the distance from that point to the focus is equal to the distance from that point to the directrix.
03

Applications and Properties

Parabolas can be seen in many real-world applications such as the design of car headlights, satellite dishes, and suspension bridge cables. A parabola has the property that, if it is made of a reflective material, all rays of light that enter it parallel to its axis of symmetry are reflected to its focus, regardless of where on the parabola they hit.

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Most popular questions from this chapter

Graph each ellipse and give the location of its foci. $$\frac{(x+3)^{2}}{9}+(y-2)^{2}=1$$

A satellite dish, like the one shown at the top of the next column, is in the shape of a parabolic surface. Signals coming from a satellite strike the surface of the dish and are reflected to the focus, where the receiver is located. The satellite dish shown has a diameter of 12 feet and a depth of 2 feet. How far from the base of the dish should the receiver be placed?

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$\frac{x^{2}}{9}+\frac{y^{2}}{36}=1$$

Graph \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) and \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\) in the same viewing rectangle for values of \(a^{2}\) and \(b^{2}\) of your choice. Describe the relationship between the two graphs.

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