Derive the equation of a parabola in the standard form y² = 4ax with diagram.

To derive the equation of a parabola in the standard form \(y^2 = 4ax\), we start by understanding the geometric properties of a parabola. A parabola is defined as the set of all points that are equidistant from a fixed point called the focus and a fixed line known as the directrix.

Key Components of the Parabola

• Focus: The point (a, 0) where the parabola opens towards.
• Directrix: The line x = -a, which is perpendicular to the axis of symmetry.
• Vertex: The midpoint between the focus and directrix, located at the origin (0, 0).

Deriving the Equation

To derive the equation, consider a point (x, y) on the parabola. The distance from this point to the focus (a, 0) is:

Distance to focus = √((x - a)² + y²)

Next, the distance from the point (x, y) to the directrix x = -a is:

Distance to directrix = x + a

Since these distances are equal for any point on the parabola, we can set them equal to each other:

√((x - a)² + y²) = x + a

Squaring Both Sides

To eliminate the square root, we square both sides:

(x - a)² + y² = (x + a)²

Expanding Both Sides

Expanding gives:

x² - 2ax + a² + y² = x² + 2ax + a²

Simplifying the Equation

Now, we can simplify by canceling out x² and a² from both sides:

-2ax + y² = 2ax

Rearranging terms leads us to:

y² = 4ax

Visual Representation

Below is a simple diagram illustrating the parabola:

      |
      |         *
      |       *   *
      |     *       *
      |   *           *
      | *               *
      |-------------------- 

This diagram shows the parabola opening to the right, with the focus at (a, 0) and the directrix at x = -a.

In summary, the standard form of the parabola \(y^2 = 4ax\) describes a curve that opens horizontally, with the vertex at the origin and the focus located at (a, 0).