Problem 24 - 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 parabola satisfying the given conditions is \(y=-x^2/60\).

Step 1: Determine the vertex

For a parabola, the vertex is the midpoint between the focus and directrix. In this case, the focus is at (0, -15) and the directrix is \(y = 15\). The y-coordinate of the vertex is thus \([-15 + 15]/2\), which evaluates to 0. Our parabola also opens downwards as the focus is below the directrix, so the vertex will be at (0, 0).

Step 2: Calculate 'p' and 'a'

The value 'p' is the distance from the vertex to the focus. Here, the vertex is at (0, 0) and the focus is at (0, -15) which means 'p' equals -15. The value 'a' is \(1/(4p)\), when calculated it results in -1/60.

Step 3: Insert values into the standard equation

Given the values of the vertex and 'a', substitute them into the standard equation for a vertically-aligned parabola, \(y = a(x - h)^2 + k\). This gives us the equation \(y = -1/60(x - 0)^2 + 0\). If we simplify this, we get \(y=-(x^2)/60\).