To derive the equation of the parabola given a focus and a directrix, we start with the focus at the point (2, 4) and the directrix as the line y = 8.
1. **Understanding the basics**: A parabola is defined as the set of all points (x, y) that are equidistant from the focus and the directrix. In this case, the focus is at (2, 4) and the directrix is the horizontal line y = 8.
2. **Distance from a point to the focus**: The distance from any point (x, y) on the parabola to the focus (2, 4) can be calculated using the distance formula:
- Distance to focus:
\( d_f = \sqrt{(x – 2)^2 + (y – 4)^2} \)
3. **Distance from a point to the directrix**: The distance from the point (x, y) to the directrix y = 8 is simply the vertical distance from y to 8, which is:
- Distance to directrix:
\( d_d = |y – 8| \)
4. **Equating the distances**: For a point (x, y) to lie on the parabola, these two distances must be equal:
- \( \sqrt{(x – 2)^2 + (y – 4)^2} = |y – 8| \)
5. **Removing the square root**: Squaring both sides to eliminate the square root gives:
- \( (x – 2)^2 + (y – 4)^2 = (y – 8)^2 \)
6. **Expanding both sides**:
- Left side:
\( (x – 2)^2 + (y – 4)^2 = (x – 2)^2 + (y^2 – 8y + 16) \) - Right side:
\( (y – 8)^2 = y^2 – 16y + 64 \)
7. **Setting up the equation**:
- So, combining our results gives us:
\( (x – 2)^2 + (y^2 – 8y + 16) = (y^2 – 16y + 64) \)
8. **Simplifying**:
- Canceling out \(y^2\) from both sides leads to:
\( (x – 2)^2 + 16y – 48 = 0 \) - Rearranging gives:
\( (x – 2)^2 = -16y + 48 \) - Which can also be written as:
\( (x – 2)^2 = -16(y – 3) \)
9. **Final form**: This shows that the parabola opens downwards, with its vertex at (2, 3). Hence, the derived equation of the parabola is:
- \( (x – 2)^2 = -16(y – 3) \)