Parabola Equation Given Focus And Directrix A Step-by-Step Guide

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Hey guys, ever wondered how to find the equation of a parabola given its focus and directrix? Well, you've come to the right place! In this article, we're going to break down the process step-by-step, making it super easy to understand. We'll specifically tackle a parabola with a focus at (0, -7π) and a directrix at y = 7π. So, buckle up and let's dive into the fascinating world of parabolas!

Understanding the Basics of a Parabola

Before we jump into solving the equation, let's make sure we're all on the same page with the fundamental concepts of a parabola. A parabola is a U-shaped curve defined as the set of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix). Think of it like this: imagine a point and a line, and then picture all the points that are the same distance away from both – that's your parabola!

  • The focus is a crucial element; it's the point inside the curve that dictates the parabola's shape.
  • The directrix is a line outside the curve, and it plays an equal role in defining the parabola.
  • The vertex is the turning point of the parabola, located exactly midway between the focus and the directrix. This point is critical because it helps us understand the orientation and position of the parabola in the coordinate plane. Understanding the vertex is essential for determining the standard equation of the parabola. The line passing through the focus and perpendicular to the directrix is the axis of symmetry, dividing the parabola into two symmetrical halves.

Key Elements and Definitions

To really grasp the concept, let's define some key terms:

  • Focus: A fixed point inside the parabola.
  • Directrix: A fixed line outside the parabola.
  • Vertex: The point where the parabola changes direction; it's the midpoint between the focus and directrix.
  • Axis of Symmetry: The line that passes through the focus and vertex, dividing the parabola into two mirror-image halves.

The Significance of the Focus and Directrix

The focus and directrix are not just random points and lines; they are the very essence of what makes a parabola a parabola. The distance from any point on the parabola to the focus is always equal to the distance from that same point to the directrix. This property is the foundation for deriving the equation of a parabola, and it's what gives the curve its distinctive U-shape. Think of it as the parabola being perfectly balanced between the allure of the focus and the repulsion of the directrix!

Deriving the Parabola Equation from Focus and Directrix

Now, let's get to the heart of the matter: how do we actually find the equation of a parabola when we know its focus and directrix? The magic lies in using the very definition of a parabola – the equal-distance property we just discussed.

The Distance Formula and the Parabola

Remember the distance formula from your geometry days? It's going to be our best friend here. The distance d between two points (x₁, y₁) and (x₂, y₂) is given by:

√((x₂ - x₁)² + (y₂ - y₁)²)

Let's say we have a point (x, y) on our parabola. The distance from (x, y) to the focus (0, -7π) must be equal to the distance from (x, y) to the directrix y = 7π. We can set up an equation using the distance formula and the distance from a point to a line.

Setting Up the Equation

  1. Distance to the Focus: Using the distance formula, the distance from (x, y) to the focus (0, -7π) is: √((x - 0)² + (y - (-7π))²) = √(x² + (y + 7π)²)
  2. Distance to the Directrix: The distance from a point (x, y) to a horizontal line y = k is simply the absolute difference in their y-coordinates: |y - k|. In our case, the directrix is y = 7π, so the distance is: |y - 7π|
  3. Equating the Distances: Now, we use the definition of the parabola and set these two distances equal to each other: √(x² + (y + 7π)²) = |y - 7π|

Solving for the Equation

Okay, we've got our equation! Now it's time to roll up our sleeves and simplify it to get the standard form of a parabola equation. This involves a bit of algebraic manipulation, but don't worry, we'll take it step-by-step.

  1. Square Both Sides: To get rid of the square root, we square both sides of the equation: (√(x² + (y + 7π)²))² = (|y - 7π|)² This simplifies to: x² + (y + 7π)² = (y - 7π)²
  2. Expand the Squares: Let's expand the squared terms: x² + (y² + 14πy + 49π²) = (y² - 14πy + 49π²)
  3. Simplify the Equation: Notice that y² and 49π² appear on both sides, so we can cancel them out: x² + y² + 14πy + 49π² = y² - 14πy + 49π² Subtracting y² and 49π² from both sides gives: x² + 14πy = -14πy
  4. Isolate the y Term: Move the -14πy term to the left side by adding 14πy to both sides: x² + 28πy = 0 Now, isolate the y term by subtracting x² from both sides: 28πy = -x²
  5. Solve for y: Finally, divide both sides by 28π to solve for y: y = -x² / (28π)

The Final Equation

And there you have it! The equation of the parabola with focus (0, -7π) and directrix y = 7π is:

y = -x² / (28π)

This equation represents a parabola that opens downwards, with its vertex at the origin (0,0). The negative sign in front of the x² term indicates that the parabola opens downwards, and the 28π in the denominator determines the