Derive the Equation of the Parabola with a Focus at (2, 4) and a Directrix of y = 8.

We will be solving this by using the focus point and the directrix.

Answer: The Equation of the Parabola with a Focus at (2, 4) and a Directrix of y = 8 is x² - 4x + 8y - 44 = 0.

Let us solve this step by step.

Explanation:

Given that, Focus at (2, 4) and a directrix of y = 8

Let us suppose that there is a point (x, y) on the parabola.

Its distance from the focus point (2, 4) is √(x - 2)² + (y - 4)² 

Its distance from directrix y = 8 is |y - 8|

Therefore, the equation will be:

√(x - 2)² + (y - 4)² = |y - 8|

Apply square on both sides.

(x - 2)² + (y - 4)² = (y - 8)² 

x² - 4x + 4 + y² - 8y + 16 = y² - 16y + 64

x² - 4x + 8y - 44 = 0

Hence, The equation of the parabola with a focus at (2, 4) and a directrix of y = 8 is x² - 4x + 8y - 44 = 0.