Parabola Equation: Focus (-5, 5), Directrix Y = -1

Oct 11, 2025 by ADMIN 51 views

Hey guys! Let's dive into deriving the equation of a parabola given its focus and directrix. This is a classic problem in mathematics, and understanding the steps involved will help you tackle similar problems with confidence. We're given a parabola with a focus at (-5, 5) and a directrix at y = -1. Our mission is to find the equation that represents this parabola. So, grab your thinking caps, and let’s get started!

Understanding the Parabola

First, let’s refresh our understanding of what a parabola actually is. At its core, a parabola is defined as the set of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix). This definition is key to deriving the equation. Imagine a point moving in a plane such that its distance from the focus always equals its distance from the directrix – the path it traces is a parabola. Now, let’s break down the key components:

Focus: The focus is a fixed point inside the curve of the parabola. It plays a crucial role in defining the shape and orientation of the parabola. Think of it as the heart of the parabola – all points on the curve are "attracted" to it.
Directrix: The directrix is a fixed line outside the curve of the parabola. It's like a guide rail for the parabola, helping to shape its curve. The directrix is always perpendicular to the axis of symmetry of the parabola.
Vertex: The vertex is the turning point of the parabola. It’s the point on the parabola that’s closest to both the focus and the directrix. It lies exactly midway between the focus and the directrix.
Axis of Symmetry: This is a line that passes through the focus and the vertex, dividing the parabola into two symmetrical halves. It’s like a mirror reflecting one side of the parabola onto the other.

With these definitions in mind, we can now approach the problem with a clear understanding of the geometry involved. Remember, the distance from any point on the parabola to the focus is equal to its distance to the directrix. This is the golden rule we'll use to derive the equation.

Setting up the Distance Formula

Let's consider a general point (x, y) on the parabola. Our goal is to express the condition that the distance from (x, y) to the focus (-5, 5) is equal to the distance from (x, y) to the directrix y = -1. To do this, we'll use the distance formula and the formula for the distance between a point and a line.

Distance to the Focus

The distance between the point (x, y) and the focus (-5, 5) can be calculated using the distance formula:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

In our case, (x₁, y₁) = (x, y) and (x₂, y₂) = (-5, 5). Plugging these values into the formula, we get:

Distance to focus = √[(x - (-5))² + (y - 5)²] = √[(x + 5)² + (y - 5)²]

This expression represents the distance from any point (x, y) on the parabola to the focus (-5, 5). We'll use this later when we set up our equation.

Distance to the Directrix

The distance between a point (x, y) and a horizontal line y = c is simply the absolute difference in their y-coordinates: |y - c|. In our case, the directrix is y = -1, so the distance from (x, y) to the directrix is:

Distance to directrix = |y - (-1)| = |y + 1|

This expression represents the distance from any point (x, y) on the parabola to the directrix y = -1. Notice that we use the absolute value because distance is always a non-negative quantity.

Now that we have expressions for both distances, we can set them equal to each other, as per the definition of a parabola. This will give us the foundation for deriving the equation.

Deriving the Equation

As we discussed earlier, the fundamental property of a parabola is that the distance from any point on it to the focus is equal to the distance from that point to the directrix. We've already calculated these distances, so let's set them equal:

√[(x + 5)² + (y - 5)²] = |y + 1|

This equation represents the parabola, but it's not in a very user-friendly form. To simplify it, we need to get rid of the square root and the absolute value. Here's how we'll do it:

Square both sides: Squaring both sides of the equation eliminates the square root. This gives us:

(x + 5)² + (y - 5)² = (y + 1)²
Expand the terms: Expand the squared terms on both sides of the equation:

(x² + 10x + 25) + (y² - 10y + 25) = (y² + 2y + 1)
Simplify the equation: Notice that the y² terms appear on both sides, so they cancel each other out. Now, let's simplify the equation by combining like terms and moving all terms to one side:

x² + 10x + 25 + y² - 10y + 25 = y² + 2y + 1 x² + 10x + 50 - 10y = 2y + 1 x² + 10x + 49 = 12y
Isolate y: To express the equation in the form y = f(x), we need to isolate y. Divide both sides by 12:

y = (1/12)(x² + 10x + 49)
Complete the square (optional): We can rewrite the equation in vertex form by completing the square. This will make it easier to identify the vertex of the parabola. To complete the square for the quadratic expression x² + 10x + 49, we need to add and subtract (10/2)² = 25 inside the parentheses:

y = (1/12)(x² + 10x + 25 + 49 - 25) y = (1/12)((x + 5)² + 24) y = (1/12)(x + 5)² + 2

So, the equation of the parabola in vertex form is:

f(x) = (1/12)(x + 5)² + 2

Analyzing the Result

We've successfully derived the equation of the parabola! Let's take a closer look at what this equation tells us about the parabola:

Vertex Form: The equation f(x) = (1/12)(x + 5)² + 2 is in vertex form, which is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. In our case, the vertex is (-5, 2). This confirms our earlier understanding that the vertex lies midway between the focus (-5, 5) and the directrix y = -1.
Direction of Opening: The coefficient 'a' in the vertex form determines whether the parabola opens upwards or downwards. Since a = 1/12, which is positive, the parabola opens upwards. This makes sense because the focus is above the directrix.
Width of the Parabola: The magnitude of 'a' also affects the width of the parabola. A smaller value of 'a' results in a wider parabola, while a larger value results in a narrower parabola. Our value of a = 1/12 indicates a relatively wide parabola.

Conclusion

Deriving the equation of a parabola from its focus and directrix is a great exercise in understanding the fundamental properties of parabolas. By using the distance formula and the definition of a parabola, we were able to arrive at the equation f(x) = (1/12)(x + 5)² + 2. This equation not only represents the parabola but also provides valuable information about its vertex, direction of opening, and width.

I hope this step-by-step guide has helped you understand the process. Remember, the key is to break down the problem into smaller, manageable steps and to rely on the fundamental definitions and formulas. Now you're equipped to tackle similar problems with confidence. Keep practicing, and you'll become a parabola pro in no time!