Understanding the Equation of a Parabola with Vertex 20 and Focus 50

Introduction to Parabolas

A parabola is a set of points in a plane that are equidistant from a fixed point (called the focus) and a fixed straight line (called the directrix). The vertex of a parabola is the point where the curve changes direction most sharply.

Deriving the Equation of a Parabola

By definition, in a parabola with focus at (F) and vertex at (V), any point (P) on the parabola satisfies the equation:

[PF text{dist}(P, L) ]

where (L) is the line perpendicular to (VF) such that (VF text{dist}(VL)).

Given Data

In our case, the vertex (V) is at 20 and the focus (F) is at 50. Therefore, the distance between the vertex and the focus, (VF), is:

[VF 50 - 20 30]

Deriving the Directrix Line

The directrix (L) is a vertical line perpendicular to the line (VF). The slope of (VF) is:

[m_{VF} frac{50 - 20}{1 - 0} 30]

The line (L) is perpendicular to (VF), so its slope is the negative reciprocal of 30, which is (-frac{1}{30}). Since the directrix is vertical, its equation is:

[x -1]

Using the Definition to Derive the Equation

For any point (P(x, y)) on the parabola, the distance from (P) to the focus (F) is:

[PF sqrt{(x - 50)^2 y^2}]

The distance from (P) to the directrix (L) is the perpendicular distance, which is the distance from (x) to (-1):

[text{dist}(P, L) |x - (-1)| |x 1|]

Therefore, setting (PF) equal to (text{dist}(P, L)), we get:

[sqrt{(x - 50)^2 y^2} |x 1|]

Squaring both sides, we obtain:

[(x - 50)^2 y^2 (x 1)^2]

Expanding both sides:

[x^2 - 10 2500 y^2 x^2 2x 1]

Subtracting (x^2) from both sides and simplifying:

[-10 2500 y^2 2x 1]

[y^2 102x - 2499]

This is the equation of the parabola in standard form:

[y^2 102x - 2499]

Conclusion

The equation of a parabola with vertex at 20 and focus at 50 is (y^2 102x - 2499). This equation describes all points (P(x, y)) that are equidistant from the focus and the directrix, fulfilling the geometric definition of a parabola.

Keywords: parabola equation, vertex, focus, geometry