To determine the equation of a parabola given its focus and directrix, we can use the geometric definition of a parabola: a set of points equidistant from the focus and the directrix.
In this case, the focus is at the point (40, 0) and the directrix is the vertical line x = 4. The parabola opens horizontally since the focus and directrix are aligned vertically.
Let the point on the parabola be represented as (x, y). The distance from this point to the focus (40, 0) is:
Distance to focus: √((x – 40)² + (y – 0)²) = √((x – 40)² + y²)
Next, we calculate the distance from the point (x, y) to the directrix (x = 4). The distance from a point to a vertical line is simply the horizontal distance:
Distance to directrix: |x – 4|
According to the definition of a parabola, these two distances must be equal:
Setting the distances equal:
√((x – 40)² + y²) = |x – 4|
We can square both sides to eliminate the square root:
(x – 40)² + y² = (x – 4)²
Expanding both sides:
(x² – 80x + 1600 + y²) = (x² – 8x + 16)
Simplifying this, we can cancel the x² terms:
-80x + 1600 + y² = -8x + 16
Bringing like terms together gives:
y² = 72x – 1584
Thus, the equation of the parabola in standard form is:
Final Equation:
y² = 72(x – 22)
In conclusion, the equation representing the parabola with the focus at (40, 0) and directrix at x = 4 is:
y² = 72(x - 22)