Introduction to Finding the Equation of a Parabola Without Calculus

The equation of a parabola can be determined if you know its focus and directrix without resorting to calculus. Fascinatingly, a parabola can be defined as the set of points whose distances to a fixed point (focus) are equal to the distances to a fixed line (directrix). This article will guide you through the process of finding the equation of a parabola using only the focus and directrix.

General Method Using Distance Formula

To find the equation of a parabola given its focus at ((f, g)) and directrix (ax by c 0), we start with the definition of a parabola. The distance from any point ((x, y)) on the parabola to the focus is equal to the distance from that point to the directrix:

[sqrt{(x - f)^2 (y - g)^2} frac{|ax by c|}{sqrt{a^2 b^2}}]

Squaring both sides of the equation eliminates the square root:

[(x - f)^2 (y - g)^2 left(frac{ax by c}{sqrt{a^2 b^2}}right)^2]

Expanding and simplifying, we get:

[b^2x^2 - 2abxy - a^2y^2 - 2(a^2b^2)facx - 2(a^2b^2)gbcy [a^2b^2f^2 g^2 - c^2] 0]

As an example, suppose that the focus is ((2, 3)) and the directrix is (5x - 7y 2 0). Here, (a 5), (b -7), and (c 2); substituting these values in, we have:

[49x^2 - 7y - 49y^2 - 14 196y 958 0]

Horizontal Directrix Cases

When the directrix is a horizontal line, say (y k), the standard form of the parabola's equation can be derived using the vertex form:

[y a(x - h)^2 k]

Here, ((h, k)) is the vertex of the parabola. If the vertex is at ((h, k)) and the parabola opens upward or downward, the formula for (a) is:

[a frac{1}{4p}]

Where (p) is the distance from the vertex to the focus (positive if the parabola opens up, negative if it opens down).

Example with Horizontal Directrix

For instance, if the vertex is ((2, 4)) and the focus is ((2, 7)), then (p 7 - 4 3). Therefore:

[a frac{1}{4 times 3} frac{1}{12}]

Hence, the equation of the parabola is:

[y frac{1}{12}(x - 2)^2 4]

Vertical Directrix Cases

If the directrix is a vertical line, follow a similar but slightly adjusted process based on the standard form of the parabola with a vertical directrix.

Non-Horizontal Directrix Cases

For parabolas with a directrix not parallel to the axes, trigonometric methods can be employed, but calculus is not necessary.

Conclusion

By using the method described above, you can find the equation of a parabola using only the coordinates of its focus and the equation of its directrix without invoking calculus. Understanding these steps and practicing with different coordinates and equations is a great way to sharpen your skills in geometric algebra.