Equation Of A Parabola With Focus (1, 2) And Directrix Y = 6

Hey guys! Today, we're diving into the fascinating world of parabolas and tackling a specific problem: finding the equation of a parabola given its focus and directrix. Specifically, we want to find the equation of a parabola with a focus at the point (1, 2) and a directrix defined by the line y = 6. Sounds intriguing, right? Let's break it down step by step.

Understanding the Basics: What is a Parabola?

Before we jump into the nitty-gritty details, let's quickly recap what a parabola actually is. In simple terms, 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. A parabola is the path traced by a point that moves in such a way that its distance to the focus is always the same as its distance to the directrix. This fundamental property is key to understanding and deriving the equation of a parabola.

The focus is a crucial element of the parabola. It's a fixed point that lies inside the curve of the parabola. The directrix, on the other hand, is a fixed line that lies outside the curve. The vertex, which is the turning point of the parabola, sits exactly halfway between the focus and the directrix. The axis of symmetry is the line that passes through the focus and the vertex, dividing the parabola into two symmetrical halves. Visualizing these components is essential for grasping the geometry of a parabola. The relationship between the focus, directrix, and vertex dictates the shape and orientation of the parabola. This definition ensures that every point on the parabola maintains an equal distance to both the focus and the directrix, a property that's fundamental to its unique shape and characteristics. Understanding this relationship is crucial for deriving the equation of a parabola and solving related problems.

Key Concepts: Focus, Directrix, and Vertex

To really nail this, let's define some important terms:

• Focus: As we mentioned, this is a fixed point inside the parabola. In our case, the focus is at (1, 2).
• Directrix: This is a fixed line outside the parabola. Here, the directrix is the line y = 6.
• Vertex: The vertex is the turning point of the parabola. It's the point where the parabola changes direction. The vertex is always exactly halfway between the focus and the directrix. Finding the vertex is a crucial step in determining the equation of the parabola. It serves as the reference point from which we measure the parabola's curvature and position in the coordinate plane. The vertex, along with the focus and directrix, forms the fundamental framework for understanding the parabola's geometry and deriving its equation. Its location dictates the parabola's orientation and the direction in which it opens.
• Axis of Symmetry: This is the line that passes through the focus and the vertex, dividing the parabola into two symmetrical halves. It's like a mirror that reflects one side of the parabola onto the other. The axis of symmetry is always perpendicular to the directrix. It provides a line of reference for understanding the parabola's symmetry and orientation. It helps us visualize how the parabola is balanced around its central axis and how its shape is reflected across this line. The axis of symmetry simplifies the analysis of the parabola's properties and aids in its graphical representation.

Finding the Vertex: The Midpoint is Key

The first thing we need to do is find the vertex. Remember, the vertex is the midpoint between the focus and the directrix. Since our focus is at (1, 2) and the directrix is y = 6, we can visualize this. The x-coordinate of the vertex will be the same as the x-coordinate of the focus, which is 1. To find the y-coordinate of the vertex, we need to find the midpoint between the y-coordinate of the focus (2) and the y-value of the directrix (6).

Think of it like finding the average: (2 + 6) / 2 = 4. So, the vertex is at the point (1, 4). This is a crucial step because the vertex gives us the center point of our parabola. Knowing the vertex allows us to set up the standard form of the parabola's equation and plug in the relevant values. The vertex, in essence, anchors the parabola in the coordinate plane, providing a reference point for all its other features and properties. Its location is essential for understanding the parabola's orientation and curvature. The vertex is not just a point; it's a fundamental element that shapes the entire parabola.

Determining the Parabola's Orientation

Next, we need to figure out which way the parabola opens. Since the directrix (y = 6) is a horizontal line above the focus (1, 2), the parabola opens downwards. Imagine the parabola