We know that a parabola is defined as the set of all points that are equidistant from the focus and the directrix. Let (x, y) be an arbitrary point on the parabola. Then, the distance from the point to the focus is given by:
√[(x - 4)² + (y + 2)²]
The distance from the point to the directrix is given by the absolute value of the difference between the y-coordinates:
| y - (-3) | = | y + 3 |
Since the point is equidistant from the focus and directrix, we can set these distances equal to each other:
√[(x - 4)² + (y + 2)²] = | y + 3 |
We can square both sides to eliminate the absolute value:
(x - 4)² + (y + 2)² = (y + 3)²
Expanding the right side and simplifying:
x² - 8x + 16 + y² + 4y + 4 = y² + 6y + 9
Simplifying further and collecting like terms:
x² - 8x + y² + 4y = -11
Completing the square for the x and y terms, we add and subtract the square of half the coefficient of x and y respectively:
(x - 4)² - 16 + (y + 2)² - 4 = -11
(x - 4)² + (y + 2)² = 9
Therefore, the equation of the parabola with focus (4, -2) and directrix y = -3 is:
(x - 4)² + (y + 2)² = 9.