Problem 21 - College Algebra

Question

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((0,15) ;\) Directrix: \(y=-15\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(x^2 = 0\).

Step 1: Find the vertex

First, it's necessary to find the vertex of the parabola, which is halfway between the focus and directrix. Since the focus is at (0,15) and the directrix is at y=-15, take the average of the y-values to find the y-coordinate of the vertex. It follows that k = (15-(-15))/2 = 15. The x-coordinate of the vertex is the same as the x-coordinate of the focus, which is 0. So the vertex is (0,15).

Step 2: Find p

Next, find the value of p, which is the distance between the vertex and the focus (or the vertex and the directrix). This can be found by subtracting the y-coordinate of the vertex from the y-coordinate of the focus (or adding the y-coordinate of the vertex to the negation of the y-coordinate of the directrix), giving p = 15 - 15 = 0 or p = 15 - (-15) = 30. Since the parabola opens upwards, p is positive.

Step 3: Plug the values into the standard form

Now that the values for the vertex (h,k) and p have been identified, they can be substituted into the standard form of the equation for a parabola, resulting in \((x - 0)^2 = 4 * 0 * (y - 15)\). This simplifies to \(x^2 = 0\).