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

Solution:

Focus at (6, 2) and a directrix of y = 1

Consider (x, y) as the point on the parabola

Distance from focus (6, 2) is √(x - 6)² + (y - 2)²

Distance from directrix y = 1 is |y - 1|

Equation of the parabola will be √(x - 6)² + (y - 2)² = |y - 1|

By squaring on both sides,

(x - 6)² + (y - 2)² = (y - 1)²

Expanding using the formulas

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

So we get,

x² - 12x - 2y + 39 = 0

Therefore, the equation of the parabola is x² - 12x - 2y + 39 = 0.

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

Summary:

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