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Chapter 10: Problem 1

Give the property that defines all parabolas.

Short Answer

Expert verified
Question: Provide a brief definition of a parabola and explain the roles of the focus and directrix in its formation. Answer: A parabola is defined as the set of all points in a plane that are equidistant from a fixed point called the focus and a fixed straight line called the directrix. The focus is a point within the parabola's interior, while the directrix is a straight line outside the parabola. The parabola's shape is determined by the symmetrical relationship between all points on the curve and their equal distance from the focus and directrix.

Step by step solution

01

Property Definition

A parabola is defined as the set of all points in a plane that are equidistant from a fixed point (called the focus) and a fixed straight line (called the directrix).
02

Understanding the Focus

In a parabola, the focus is a fixed point that lies within the parabola's interior. The distance between any point on the parabola and the focus is called the focal length.
03

Understanding the Directrix

The directrix is a fixed straight line that lies outside the parabola. Every point on the parabola is equidistant to the focus and the directrix.
04

The Parabola Equation

For a parabola with a vertical axis of symmetry and focus at (h, k + p) and directrix y = k - p, we can write its equation as: \[ (x - h)^2 = 4p(y - k) \] To find the equation for a parabola with a horizontal axis of symmetry and focus at (h + p, k) and directrix x = h - p, we can use the equation: \[ (y - k)^2 = 4p(x - h) \]
05

Visualizing a Parabola

Imagine a graph with a focus at point F and a directrix as a straight line parallel to the x-axis or y-axis. The parabola's shape will be a curve where all points on the parabola mirror each other across the vertical or horizontal line of symmetry. The parabola is symmetric and ordinarily shaped like a U or an inverted U (for vertical parabolas), or shaped like an opening to the right or to the left for (horizontal parabolas).

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