Finding The Equation Of A Parabola With Focus (0,-6) And Directrix Y=6

Understanding parabolas is crucial in mathematics, especially in analytic geometry. A parabola is defined as the set of all points equidistant to a fixed point (the focus) and a fixed line (the directrix). In this comprehensive guide, we will walk through the process of finding the equation of a parabola, focusing on a specific example where the focus is at the point (0, -6) and the directrix is the line y = 6. This exploration will not only help you solve this particular problem but also equip you with the skills to tackle similar challenges involving parabolas.

Understanding the Definition of a Parabola

Before diving into the solution, it's essential to grasp the fundamental definition of a parabola. A parabola is a conic section formed by the intersection of a right circular cone and a plane parallel to a generating straight line of the cone. More simply, a parabola is the set of all points in a plane that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix. The line perpendicular to the directrix and passing through the focus is the axis of symmetry of the parabola. The point where the parabola intersects its axis of symmetry is called the vertex. This definition is the cornerstone of deriving the equation of a parabola, and understanding it thoroughly will make the process much clearer.

Determining the Vertex of the Parabola

The vertex of a parabola is the midpoint between the focus and the directrix. It's a crucial point because it helps us determine the standard form of the parabola's equation. In our case, the focus is at (0, -6) and the directrix is the line y = 6. To find the vertex, we need to find the midpoint between the focus and the directrix. Since the focus has coordinates (0, -6), its y-coordinate is -6. The directrix is the line y = 6, so any point on the directrix has a y-coordinate of 6. The midpoint’s y-coordinate is the average of these two values: (-6 + 6) / 2 = 0. The x-coordinate of the focus is 0, and since the directrix is a horizontal line, the x-coordinate of the vertex will also be 0. Therefore, the vertex of the parabola is at the point (0, 0). This means our parabola is centered at the origin, which simplifies our calculations.

Using the Distance Formula

The definition of a parabola states that for any point (x, y) on the parabola, the distance to the focus is equal to the distance to the directrix. We can use the distance formula to express these distances. The distance d between two points (x₁, y₁) and (x₂, y₂) in a plane is given by the formula:

$\qquad d = \sqrt{(x₂ - x₁)² + (y₂ - y₁)²}$

Let's apply this to our problem. Let (x, y) be any point on the parabola. The distance from (x, y) to the focus (0, -6) is:

$\qquad d_\text{focus} = \sqrt{(x - 0)² + (y - (-6))²} = \sqrt{x² + (y + 6)²}$

The distance from (x, y) to the directrix y = 6 is the perpendicular distance, which is simply the absolute difference in the y-coordinates:

$\qquad d_\text{directrix} = |y - 6|$

According to the definition of a parabola, these two distances must be equal:

$\qquad \sqrt{x² + (y + 6)²} = |y - 6|$

Deriving the Equation of the Parabola

Now that we have the equation equating the distances, we need to simplify it to find the standard form of the parabola's equation. We start by squaring both sides of the equation to eliminate the square root and the absolute value:

$\qquad (\sqrt{x² + (y + 6)²})² = (|y - 6|)²$

This simplifies to:

$\qquad x² + (y + 6)² = (y - 6)²$

Next, we expand the squared terms:

$\qquad x² + (y² + 12y + 36) = y² - 12y + 36$

Now, we can simplify by canceling out the y² and 36 terms on both sides:

$\qquad x² + 12y = -12y$

Add 12y to both sides to isolate the y terms:

$\qquad x² = -24y$

Finally, solve for y to get the equation in the standard form:

$\qquad y = -\frac{1}{24}x²$

This is the equation of the parabola with focus (0, -6) and directrix y = 6. The negative coefficient indicates that the parabola opens downwards.

Identifying the Correct Option

By deriving the equation of the parabola, we found that $y = -\frac{1}{24}x²$. Now, we need to match this equation with the given options. The options are:

a. $y = -\frac{1}{6}x²$

b. $y = \frac{1}{24}x²$

c. $y = \frac{1}{6}x²$

d. $y = -\frac{1}{24}x²$

Comparing our derived equation with the options, we can see that option (d) matches exactly. Therefore, the correct equation of the parabola is:

$\qquad y = -\frac{1}{24}x²$

Conclusion

In this detailed guide, we successfully found the equation of a parabola with a focus of (0, -6) and a directrix of y = 6. We started by understanding the definition of a parabola and its key components, such as the focus, directrix, and vertex. We then used the distance formula to express the condition that any point on the parabola is equidistant from the focus and the directrix. By setting these distances equal and simplifying the resulting equation, we derived the equation of the parabola in the form $y = -\frac{1}{24}x²$. Finally, we compared our result with the given options and identified the correct answer.

This process demonstrates a systematic approach to solving problems involving parabolas. By understanding the fundamental principles and applying the appropriate formulas, you can confidently tackle a wide range of parabola-related problems. Remember, the key is to break down the problem into manageable steps and carefully apply the definitions and formulas.

Practice Problems

To further solidify your understanding, try solving similar problems. For example:

1. Find the equation of a parabola with a focus of (0, 6) and directrix y = -6.
2. Determine the equation of a parabola with a focus of (2, 0) and directrix x = -2.
3. What is the equation of a parabola with a focus of (-3, 4) and directrix y = 2?

By working through these practice problems, you will reinforce your skills and gain a deeper understanding of parabolas and their equations.