Deriving the Equation of a Parabola with an Axis Parallel to the X-Axis

In this article, we will explore the process of finding the equation of a parabola whose axis is parallel to the x-axis. This is a common problem in analytic geometry, and it requires us to understand the standard form of a parabola and apply it to the given conditions.

Equation of Parabola with Axis Parallel to X-Axis

The standard form of a parabola with its axis parallel to the x-axis is given by:

y - k2 4a(x - h)

Given Conditions and Initial Equations

We are given three points that the parabola passes through:

(1.5, 50) (-1, 50) (-1, -12)

This information allows us to set up a system of equations based on the standard form of the parabola.

Setting Up the Equations

First, we substitute the first point (1.5, 50) into the equation:

50 - k2 4a(1.5 - h)

Which simplifies to:

1 - k2 4a(1.5 - h)

Substitute the second point (-1, 50) into the equation:

50 - k2 4a(-1 - h)

Which simplifies to:

k2 4a(-1 - h)

Substitute the third point (-1, -12) into the equation:

-12 - k2 4a(-1 - h)

Which simplifies to:

2 - k2 4a(-1 - h)

Solving for a and h

Now, let's solve these simultaneous equations. Subtract the second equation from the first:

1 - k2 - k2 4a(1.5 - h) - 4ak(-1 - h)

This simplifies to:

-2k 6a - 20a

We can further simplify this to:

-14a -2k

Thus:

a k/7

Next, subtract the third equation from the second:

k2 - (-12 - k2) 4a(-1 - h) - 4ak(-1 - h)

This simplifies to:

24k 20a

We already know that a k/7, so substitute this back:

24k 20 * k/7

Which simplifies to:

24 20/7

Now, multiply both sides by 7:

168 20

Multiply both sides by 7 again:

2k - 2 12a

Substitute a k/7:

2k - 2 12 * k/7

This simplifies to:

2k - 2 12k/7

Subtract 12k/7 from both sides:

14k - 12k 14

Simplify:

2k 14

Divide both sides by 2:

k 7

Now, substitute k 7 back into a k/7:

a 7/7 1

Finding the Vertex and Additional Values

Next, we substitute k 7 into the simplified form of the equation to find the vertex:

1 - 72 4(1)(2)(x - h)

This simplifies to:

1 - 49 8(x - h)

Which further simplifies to:

-48 8(x - h)

Divide both sides by 8:

-6 x - h

Substitute the point (1.5, 50) into the equation:

1.5 -6 h

Solving for h:

h 7.5

Equation of the Parabola

Substitute a 1, k 7, and h 7.5 into the standard form:

y - 72 4(1)(x - 7.5)

This simplifies to:

y - 49 4(x - 7.5)

Additional Information

The vertex of the parabola is at (7.5, 49).

The distance between the focus (F) and the vertex (V) is:

FA 1.5

The directrix is at:

x -6.5

Deriving Parabola Equations in General

For a parabola with its axis parallel to the x-axis, the general form is:

y - β2 k(x - α)

Where (α, β) is the vertex of the parabola and k is the length of the latus rectum.

Alternative Methods

To find the equation of a parabola using the points it passes through, you can use the general form:

x ay2 by c

By substituting the coordinates of the three given points, you can solve for the unknowns a, b, and c.

Conclusion

Understanding the process of deriving the equation of a parabola with its axis parallel to the x-axis is essential for solving geometric problems. This method involves setting up and solving a system of equations based on given conditions. The derived equation and additional information provide a comprehensive understanding of the parabola's properties.