Parabola (Focus-Directrix Definition) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Parabola (Focus-Directrix Definition).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

Every point on a parabola is exactly the same distance from the focus as it is from the directrix line. This geometric property is why satellite dishes and flashlight reflectors are parabolic—signals from the focus reflect off the curve in parallel lines.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The focus-directrix definition reveals that a parabola is a conic section with eccentricity e = 1—exactly between an ellipse (e < 1) and a hyperbola (e > 1).

Common stuck point: The vertex is halfway between the focus and directrix. If you know any two of {vertex, focus, directrix}, you can find the third.

Sense of Study hint: Find p first: the distance from the vertex to the focus. Then the directrix is the same distance p on the opposite side of the vertex.

Worked Examples

Example 1

easy
Find the focus and directrix of the parabola y = \frac{1}{8}x^2.

Solution

  1. 1
    Rewrite in standard form: x^2 = 8y. This matches x^2 = 4py where 4p = 8, so p = 2.
  2. 2
    The parabola opens upward. The focus is at (0, p) = (0, 2).
  3. 3
    The directrix is y = -p = -2.

Answer

\text{Focus: } (0, 2); \quad \text{Directrix: } y = -2
For a parabola x^2 = 4py, the parameter p is the distance from the vertex to the focus (and also from the vertex to the directrix). If p > 0, the parabola opens upward; if p < 0, it opens downward.

Example 2

medium
Write the equation of a parabola with vertex at the origin, opening to the right, with focus at (3, 0).

Example 3

medium
Find the focus and directrix of the parabola y = \frac{1}{8}x^2.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Find the equation of the parabola with focus (0, -4) and directrix y = 4.

Example 2

hard
A parabola has vertex (2, -1) and focus (2, 3). Find its equation and the length of the latus rectum.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

quadratic functions