Deriving the Equation of a Parabola with Focus and Directrix

In this detailed guide, we will derive the equation of a parabola given its focus and directrix, using the geometric definition of a parabola. We will explore the process step-by-step, ensuring clarity and understanding for both students and SEO enthusiasts.

Introduction to Parabolas

A parabola is the set of all points in a plane that are equidistant from a fixed point, known as the focus, and a fixed line, known as the directrix. This definition forms the basis for deriving the equation of a parabola.

Given Information

For this specific problem, we are given:

Focus: (F(-1, -1)) Directrix: (y -x)

Calculating the Distance from a Point to the Focus

The distance from a point ((x, y)) to the focus ((-1, -1)) is given by:

[d_F sqrt{(x 1)^2 (y 1)^2}]

Calculating the Distance from a Point to the Directrix

The distance from a point ((x, y)) to the line (y -x) can be calculated using the formula for the distance from a point to a line (Ax By C 0). The line (y -x) can be rewritten as:

[x y 0 quad text{(where } A 1, B 1, text{ and } C 0)]

The distance (d_D) from the point ((x, y)) to the directrix is:

[d_D frac{|x y|}{sqrt{1^2 1^2}} frac{|x y|}{sqrt{2}}]

Setting the Distances Equal

According to the definition of a parabola, we set (d_F d_D):

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

Squaring Both Sides to Eliminate the Square Root

By squaring both sides, we eliminate the square root:

[(x 1)^2 (y 1)^2 left(frac{x y}{sqrt{2}}right)^2]

This simplifies to:

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

Multiplying Through by 2 for Clarity

Multiplying through by 2 to eliminate the fraction:

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

Expanding and Simplifying the Equation

Expanding both sides:

[2(x^2 2x 1) 2(y^2 2y 1) x^2 2xy y^2]

This results in:

[2x^2 4x 2 2y^2 4y 2 x^2 2xy y^2]

Combining like terms:

[2x^2 2y^2 4x 4y 4 x^2 2xy y^2]

Rearranging the terms:

[2x^2 2y^2 - 2xy 4x 4y 4 - x^2 - y^2 0][x^2 - 2xy y^2 4x 4y 4 0]

Final Equation of the Parabola

The equation of the parabola is:

[x^2 - 2xy y^2 4x 4y 4 0]

This equation represents the relationship between (x) and (y) given the specific focus and directrix.

Conclusion

The process of deriving the equation of a parabola using its focus and directrix is a fundamental concept in mathematics. Understanding this equation can help in various applications and further mathematical explorations.