How to Find the Equation of a Parabola Given Focus and Directrix

Understanding the relationship between the focus and directrix of a parabola is key to finding its equation. Parabolas, being the set of all points equidistant from a fixed point (the focus) and a straight line (the directrix), have a distinct standard form that reflects this relationship. This article will guide you through the process of deriving the equation of a parabola given its focus and directrix, highlighting the mathematical principles involved.

Identifying the Focus and Directrix

First, it's essential to identify the focus and directrix of the parabola. The focus can be represented by the coordinates (h, k), and the directrix can be a horizontal or vertical line. If the directrix is horizontal, it can be expressed as y d, and if it is vertical, it can be expressed as x d. The value c represents the distance from the vertex to the directrix.

Determining the Orientation of the Parabola

The orientation of the parabola—whether it opens upwards, downwards, to the right, or to the left—depends on the relative positions of the focus and the directrix. For a vertical parabola, if the directrix is a horizontal line, the parabola opens upwards if the focus is above the directrix and downwards if the focus is below. For a horizontal parabola, if the directrix is a vertical line, the parabola opens to the right if the focus is to the right of the directrix and to the left if the focus is to the left.

Using the Definition of a Parabola: Equidistant Points

A parabola is defined as the set of all points (x, y) that are equidistant from the focus and the directrix. Let's break down the process with both vertical and horizontal parabolas.

Vertical Parabolas

For a vertical parabola, the distance from a point (x, y) to the focus (h, k) is given by:

sqrtr(x - h^2 y - k^2)

The distance from the point (x, y) to the directrix y d is:

|y - d|

Setting these distances equal gives the equation to solve:

sqrtr(x - h^2 y - k^2) |y - d|

Horizontal Parabolas

For a horizontal parabola, the distance from a point (x, y) to the focus (h, k) is again given by:

sqrtr(x - h^2 y - k^2)

The distance from the point (x, y) to the directrix x d is:

|x - d|

Setting these distances equal gives the equation to solve:

sqrtr(x - h^2 y - k^2) |x - d|

Simplifying to Find the Equation of the Parabola

To find the standard form of the parabola's equation, we square both sides of the resulting equation to eliminate the square root and then simplify. This can be demonstrated with an example.

Example: Finding the Equation of a Parabola

Given:

- Focus: (2, 3)

- Directrix: y 1

Steps:

1. The directrix is a horizontal line, so the parabola opens upwards since the focus is above the directrix.

2. Set up the equation using the distance from the point (x, y) to the focus (2, 3) and the directrix y 1:

sqrtr(x - 2^2 y - 3^2) |y - 1|

3. Square both sides of the equation:

(x - 2^2 y - 3^2) (y - 1^2)

4. Expand and simplify:

x - 2^2 y^2 - 6y 9 y^2 - 2y 1

5. Simplify to get the standard form of the parabola:

x - 4 4y - 8y 8 0

x 4y - 8 0

This results in the equation of the parabola in the simplified form.

By following these detailed steps, you can derive the equation of a parabola given its focus and directrix, ensuring a thorough understanding of the geometric and algebraic relationships involved.