To derive the equation of the parabola, we start by understanding the definition of a parabola: it is the set of all points that are equidistant from a focus point and a directrix line.
In this case, the focus is at the point (6, 2) and the directrix is the line y = 1. The equation of the parabola can be found using the distance formula.
Let (x, y) be any point on the parabola. The distance from the point (x, y) to the focus (6, 2) can be calculated as:
Dfocus = √(x - 6)² + (y - 2)²
The distance from the point (x, y) to the directrix y = 1 is simply the vertical distance:
Ddirectrix = |y - 1|
Since these distances are equal, we set them equal to each other:
√(x - 6)² + (y - 2)² = |y - 1|
Next, we will square both sides to eliminate the square root, keeping in mind that we need to consider both cases of the absolute value:
(x - 6)² + (y - 2)² = (y - 1)²
Expanding both sides:
(x - 6)² + (y² - 4y + 4) = (y² - 2y + 1)
Now, we can simplify this equation:
(x - 6)² + y² - 4y + 4 = y² - 2y + 1
Subtracting y2 from both sides and simplifying:
(x - 6)² - 4y + 4 = -2y + 1
Now, we simplify further:
(x - 6)² - 2y + 3 = 0
Rearranging gives us the standard form of the parabola:
2y = (x - 6)² + 3
This can be rewritten as:
y = ½(x - 6)² + &frac32;
The final equation of the parabola with focus at (6, 2) and directrix y = 1 is:
y = ½(x - 6)² + &frac32;