To derive the equation of the parabola with a given focus and directrix, we need to follow the definition of a parabola: it is the set of all points that are equidistant from the focus and the directrix.
Here, the focus is at the point (6, 2), and the directrix is the line y = 1. The distance from any point (x, y) on the parabola to the focus (6, 2) must equal the perpendicular distance from (x, y) to the directrix, which is the line y = 1.
1. **Distance to the focus**:
The distance from a point (x, y) to the focus (6, 2) is given by:
Dfocus = √((x – 6)² + (y – 2)²)
2. **Distance to the directrix**:
The distance from the point (x, y) to the directrix (y = 1) is simply the vertical distance:
Ddirectrix = |y – 1|
3. **Setting the distances equal**:
Since both distances are equal for points on the parabola, we can set up the equation:
√((x – 6)² + (y – 2)²) = |y – 1|
4. **Squaring both sides** to eliminate the square root:
(x – 6)² + (y – 2)² = (y – 1)²
5. **Expanding both sides**:
(x – 6)² + (y² – 4y + 4) = (y² – 2y + 1)
6. **Simplifying**:
(x – 6)² + y² – 4y + 4 = y² – 2y + 1
Now, if we subtract y² from both sides:
(x – 6)² – 4y + 4 = -2y + 1
7. **Bringing all terms involving y to one side:**
(x – 6)² + 2y + 3 = 0
Finally, isolate y:
2y = -(x – 6)² – 3
y = -½(x – 6)² – ¾
Thus, the equation of the parabola with a focus at (6, 2) and a directrix of y = 1 is:
y = -½(x – 6)² – ¾