The directrix and focus of a parabola

The directrix and focus are fundamental elements in defining a parabola, a type of conic section or curve. Each parabola is symmetrical and has a single axis of symmetry. The focus and directrix work together to provide a unique way to construct or understand the shape and orientation of a parabola.

Focus:

The focus of a parabola is a fixed point located inside the curve. It has a unique property: for any point on the parabola, the distance to the focus is equal to the distance to the directrix (a straight line). The focus lies on the axis of symmetry of the parabola.

Directrix:

The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola and does not intersect the parabola itself. The distance between any point on the parabola and the directrix is equal to the distance between that point and the focus. This constant distance defines the parabola's curvature and width.

Relationship and Properties:

• Equidistance property: Every point on the parabola is equidistant from the focus and the directrix. This property is fundamental to the mathematical definition of a parabola.
• Formula and Orientation: The standard form of a parabola's equation depends on its orientation (vertical or horizontal) and its distance from the focus to the vertex (point where the parabola changes direction).

For a vertical parabola with vertex at the origin, the equation can be written as $y = ax^2$ if the focus is above the directrix, and $y = -ax^2$ if the focus is below the directrix, where $4a$ is the distance between the vertex and the focus (or the vertex and the directrix). For a horizontal parabola, the roles of $x$ and $y$ are switched.

How to find the directrix and focus of a parabola?

Finding the directrix and focus of a parabola involves using the equation of the parabola and its geometric properties. Here's how to do it step by step:

Given Equation Form:

Assume the parabola has a vertex at the origin $(0, 0)$ and is oriented either vertically (opening upwards or downwards) or horizontally (opening to the right or left). The standard equations are:

• For a vertical parabola: $y = ax^2$
• For a horizontal parabola: $x = ay^2$

where $a$ is a constant that determines the "openness" of the parabola.

Step 1: Identify the Orientation and Equation

First, determine if the parabola is vertical ($y = ax^2$) or horizontal ($x = ay^2$). This will tell you the direction in which the parabola opens.

Step 2: Determine the Value of $a$

The value of $a$ in the equation tells you how "wide" or "narrow" the parabola is and helps in finding the focus and directrix.

Finding the Focus:

• For a vertical parabola $y = ax^2$, the focus is at $(0, \frac{1}{4a})$.
• For a horizontal parabola $x = ay^2$, the focus is at $(\frac{1}{4a}, 0)$.

Finding the Directrix:

• For a vertical parabola $y = ax^2$, the directrix is a horizontal line given by $y = -\frac{1}{4a}$.
• For a horizontal parabola $x = ay^2$, the directrix is a vertical line given by $x = -\frac{1}{4a}$.

Solved Example

Examples for a Vertical Parabola:

Given: $y = 4x^2$

Find $a$:

Here, $a = 4$, so the coefficient $1/4a = 1/16$.

Focus: The focus is $(0, 1/16)$.

Directrix: The directrix is the line $y = -1/16$.

Example for a Horizontal Parabola:

Given: $x = \frac{1}{2}y^2$

Find $a$:

Here, $a = 1/2$, so $1/4a = 1/2$.

Focus:

The focus is $(1/2, 0)$.

Directrix:

The directrix is the line $x = -1/2$.

Parabola equation from focus and directrix

To derive the equation of a parabola from its focus and directrix, you can use the definition of a parabola: it is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Given:

• Focus: $F(p, q)$
• Directrix: Line with equation $ax + by + c = 0$

Objective:

Derive the equation of the parabola.

Steps:

Step 1: Use the Distance Formula

For any point $P(x, y)$ on the parabola, the distance to the focus $F(p, q)$ is equal to the perpendicular distance from $P(x, y)$ to the directrix $ax + by + c = 0$.

Distance to Focus:

$D_F = \sqrt{(x - p)^2 + (y - q)^2}$

Distance to Directrix:

$D_D = \frac{|ax + by + c|}{\sqrt{a^2 + b^2}}$

Step 2: Set Distances Equal

Because $D_F = D_D$, you have: $\sqrt{(x - p)^2 + (y - q)^2} = \frac{|ax + by + c|}{\sqrt{a^2 + b^2}}$

Squaring both sides to eliminate the square root gives: $(x - p)^2 + (y - q)^2 = \left(\frac{ax + by + c}{\sqrt{a^2 + b^2}}\right)^2$

Step 3: Simplify the Equation

Depending on the orientation and specific values of $p$, $q$, $a$, $b$, and $c$, you'll expand and simplify the equation to find the standard form of the parabola.

Example for a Vertical Parabola:

Given:

• Focus: $F(0, p)$ (assuming $p > 0$)
• Directrix: $y = -p$

Using the definition:

Distance to Focus:

For any point $P(x, y)$ on the parabola, the distance to the focus is: $D_F = \sqrt{(x - 0)^2 + (y - p)^2}$

Distance to Directrix:

The distance to the directrix $y = -p$ is simply: $D_D = |y + p|$

Setting $D_F = D_D$ and squaring both sides, we get: $x^2 + (y - p)^2 = (y + p)^2$

Expanding and then simplifying: $x^2 + y^2 - 2yp + p^2 = y^2 + 2yp + p^2$ $x^2 = 4yp$

Since the formula for a vertical parabola with vertex at the origin can be written as $y = \frac{1}{4p}x^2$, we match the derived equation with this form: $y = \frac{1}{4p}x^2$

Conclusion:

This process illustrates how to derive the equation of a parabola using the geometrical definition involving the distance of any point on the parabola from the focus and the directrix. Depending on the given focus and directrix, the approach helps in crafting the parabola's equation in its standard form.