Final answer:
The correct equation for the parabola with a focus at (0,3) and a directrix at y=5 is D) x² + (y - 3)² = 4, which follows from the definition of a parabola and equating the distance from a point to the focus and the directrix.
Step-by-step explanation:
To determine the equation of a parabola given the focus and directrix, we can use the definition of a parabola as the set of all points that are equidistant from the focus and the directrix. The focus is given as (0,3) and the directrix is the line y=5. The distance of any point (x, y) from the focus is the square root of the sum of the squares of the difference in x-coordinates and y-coordinates, i.e., √((x-0)²+(y-3)²), which simplifies to √(x²+(y-3)²). The distance of any point (x, y) from the directrix y=5 is simply the absolute value of the difference in y-coordinates, which is |y-5|. The equation of the parabola is obtained by equating these distances:
|y-5| = √(x²+(y-3)²)
Squaring both sides to remove the square root gives:
(y-5)² = x²+(y-3)²
Expanding and simplifying, we get:
y²-10y+25 = x²+y²-6y+9
After canceling y² and simplifying further, we arrive at:
x²+4y=16
Dividing everything by 4 to complete the square on y gives:
x²+(y-4)²=4
Which matches the form of an equation of a parabola with vertex form:
(x-h)²+(y-k)²=4p
where h and k are vertex coordinates and p is the distance from the vertex to the focus. Comparing this, the correct answer is:
D) x² + (y - 3)² = 4