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Directrix

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Directrix

Study Guide

Key Definition

A parabola is a set of all points in a plane which are an equal distance away from a given point called the focus and a line called the directrix. The directrix is perpendicular to the axis of symmetry of a parabola.

Important Notes

The directrix is a line that does not touch the parabola.
If the parabola opens upwards or downwards, the directrix is a horizontal line.
If the parabola opens left or right, the directrix is a vertical line.
The vertex of a parabola is equally distant from the focus and the directrix.
Equation of a directrix: horizontal $y = c$ or vertical $x = c$.
The midpoint between the focus and directrix is the vertex.

Mathematical Notation

$F(x_F, y_F)$ denotes the focus of the parabola

$D: y = c$ (or $x = c$) denotes the directrix

$V(h, k)$ denotes the vertex (midpoint of focus and directrix)

$d(P, F) = d(P, D)$ condition defining the parabola

Remember to use proper notation when solving problems

Why It Works

The directrix allows us to define a parabola by ensuring that every point is equidistant from the focus and the line. This geometric property helps in plotting and analyzing parabolas.

Remember

The vertex is always midway between the focus and directrix.

Quick Reference

Vertex Formula:$\text{Midpoint of } F \text{ and } D$

Directrix Equation:$y = c$ or $x = c$

Understanding Directrix

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A parabola is the set of all points equidistant from a fixed point (the focus) and a line (the directrix). For a horizontal directrix $y = c$, any point $(x,y)$ on the parabola satisfies $d((x,y),F) = |y - c|$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If a parabola has a vertex at $(1, 4)$ and focus at $(1, 2)$, what is the equation of the directrix?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you're designing a skate ramp which follows the path of a parabola. The vertex of the ramp is at $(0, 3)$ and the focus is at $(0, 5)$. What is the directrix?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given a parabola with vertex at $(-2, 0)$ and a directrix $y = 2$, find the focus.

Challenge Quiz

Single Choice Quiz

Advanced

For a parabola with the vertex at $(3, 1)$ and directrix $y = 5$, find the focus.

Recap

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Review key concepts and takeaways