Parabolas: Focus and Directrix Examples

Understanding Parabolas: Focus and Directrix 🧐

A parabola is defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). Let's explore how to find these key elements given the equation of a parabola.

Standard Forms of a Parabola

The standard forms of a parabola are:

• Vertical Parabola: $(x - h)^2 = 4p(y - k)$
• Horizontal Parabola: $(y - k)^2 = 4p(x - h)$

Where:

• $(h, k)$ is the vertex of the parabola.
• $p$ is the distance from the vertex to the focus and from the vertex to the directrix.

Finding the Focus and Directrix: Step-by-Step

1. Identify the Vertex: Determine $(h, k)$ from the equation.
2. Find $p$: Compare the given equation to the standard form to find the value of $p$.
3. Locate the Focus:
   • For a vertical parabola, the focus is at $(h, k + p)$.
   • For a horizontal parabola, the focus is at $(h + p, k)$.
4. Determine the Directrix:
   • For a vertical parabola, the directrix is the line $y = k - p$.
   • For a horizontal parabola, the directrix is the line $x = h - p$.

Example 1: Vertical Parabola ⬆️

Consider the equation: $(x - 2)^2 = 8(y + 1)$

1. Vertex: $(h, k) = (2, -1)$
2. Find $p$: $4p = 8$, so $p = 2$
3. Focus: $(h, k + p) = (2, -1 + 2) = (2, 1)$
4. Directrix: $y = k - p = -1 - 2 = -3$, so $y = -3$

Example 2: Horizontal Parabola ➡️

Consider the equation: $(y - 3)^2 = -12(x + 4)$

1. Vertex: $(h, k) = (-4, 3)$
2. Find $p$: $4p = -12$, so $p = -3$
3. Focus: $(h + p, k) = (-4 + (-3), 3) = (-7, 3)$
4. Directrix: $x = h - p = -4 - (-3) = -1$, so $x = -1$

Example 3: Parabola in Standard Form ✍️

Consider the equation: $y^2 = 4x$

1. Vertex: $(h, k) = (0, 0)$
2. Find $p$: $4p = 4$, so $p = 1$
3. Focus: $(h + p, k) = (0 + 1, 0) = (1, 0)$
4. Directrix: $x = h - p = 0 - 1 = -1$, so $x = -1$

Code Example (Python) 🐍

Here's a Python function to calculate the focus and directrix for a parabola in the form $(x - h)^2 = 4p(y - k)$:

def find_focus_directrix_vertical(h, k, p):
 focus = (h, k + p)
 directrix = k - p
 return focus, directrix

# Example usage:
h = 2
k = -1
p = 2
focus, directrix = find_focus_directrix_vertical(h, k, p)
print(f"Focus: {focus}")
print(f"Directrix: y = {directrix}")

Key Takeaways 🔑

• The value of $p$ determines the direction and width of the parabola.
• If $p > 0$, the parabola opens upwards (vertical) or to the right (horizontal).
• If $p < 0$, the parabola opens downwards (vertical) or to the left (horizontal).