Introduction to Parabolas: Focus and Directrix

A parabola is a conic section defined by the set of all points equidistant from a fixed point, called the focus, and a fixed line, called the directrix. This article will guide you through the process of finding the standard form of a parabola when given the focus and the directrix. Let's explore a real-world example with the given focus and directrix to understand the procedure better.

Problem Statement

Given the focus at F(7, 11) and the directrix x 1, we need to find the standard form of the parabola.

Step-by-Step Solution

Step 1: Determine the Vertex

The vertex of the parabola is located halfway between the focus and the directrix. Since the directrix is vertical, the x-coordinate of the vertex is the average of the x-coordinate of the focus and the directrix.

(x_v frac{x_F x_{directrix}}{2} frac{7 1}{2} 4)

The y-coordinate of the vertex is the same as that of the focus.

(y_v 11)

Thus, the vertex (V) is at ((4, 11)).

Step 2: Determine the Orientation

Since the focus is to the right of the directrix, the parabola opens to the right.

Step 3: Calculate (p)

The value (p) is the distance from the vertex to the focus or the vertex to the directrix. We can calculate (p) as follows:

(p x_F - x_v 7 - 4 3)

Step 4: Write the Standard Form

For a parabola that opens to the right, the standard form is given by:

(y - k^2 4p(x - h))

where ((h, k)) is the vertex. Plugging in our values:

(h 4,) (k 11,) (p 3)

Now substituting these into the equation:

(y - 11^2 4 cdot 3(x - 4))

This simplifies to:

(y - 121 12x - 48)

Final Answer

The standard form of the parabola is:

(y - 121 12x - 48)

Let's simplify this to the standard form:

(y - 121 12x - 48)

Which simplifies to:

(y - 11^2 12x - 4)

This is the required standard form of the parabola.

Verification

If ((x, y)) is a point on this parabola, its distance to the focus is (sqrt{(x - 7)^2 (y - 11)^2}). Its distance from the directrix is (x - 1). Since this is a parabola, those distances are equal:

(sqrt{(x - 7)^2 (y - 11)^2} x - 1)

Squaring both sides, we get:

((x - 7)^2 (y - 11)^2 (x - 1)^2)

Expanding and simplifying, we arrive at the standard form of the parabola:

(x frac{y^2 - 22y 169}{12})

This confirms our solution is correct.

Related Concepts

Understanding parabolas is crucial for various applications in mathematics, physics, and engineering. Other related concepts include:

Equation of a Parabola: Knowledge of the standard form, vertex form, and factored form of a parabola. Focal Diameter and Latus Rectum: The length of the latus rectum and the distance from the focus to the directrix. Vertex and Focus Calculation: Techniques for finding the vertex and focus given the equation or other parameters.

By mastering these concepts, you can solve a wide range of problems involving parabolas in real-world applications.