Derive the Equation of the Parabola with a Focus at (6, 2) and a Directrix of y = 1.

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

Answer: The equation of the parabola with a focus at (6, 2) and a directrix of y = 1 is x² - 12x - 2y + 39 = 0.

Let us solve this step by step.

Explanation:

Given that, focus at (6, 2) and a directrix of y = 1.

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

Its distance from the focus point (6, 2) is √(x - 6)² + (y - 2)²

Its distance from directrix y = 1 is |y - 1|.

Therefore, the equation will be:

√(x - 6)² + (y - 2)² = |y - 1|

Squaring both sides.

(x - 6)² + (y - 2)² = (y - 1)²

x² - 12x + 36 + y² - 4y + 4 = y² - 2y + 1

x² - 12x - 2y + 39 = 0