Problem 23 - College Algebra

Question

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((0,-25) ;\) Directrix: \(y=25\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(y = (1/100)x^2\).

Step 1: Find the Vertex of the Parabola

The vertex of the parabola is located exactly between the focus and directrix. Since the focus is at \((0,-25)\) and the directrix is \(y=25\), the y-coordinate of the vertex (\(k\)) is halfway between -25 and 25, which is 0. Also since the x-coordinate of the focus is 0, and the vertex is aligned with the focus in case of a vertical parabola, the x-coordinate of the vertex (\(h\)) is also zero. Therefore, the vertex coordinates are (0,0).

Step 2: Determine the Value of 'a'

We can find 'a' using the formula \(a = 1/(4f)\), where 'f' is the distance of the vertex from the focus. Since our vertex is (0,0) and the focus is (0, -25), 'f' equals 25. Plugging this into our formula gives \(a = 1/(4*25) = 1/100\).

Step 3: Write the Standard Form of the Equation

The standard form for this parabola is \(y = a(x - h)^2 + k\). We found that \(a = 1/100\), \(h = 0\), and \(k = 0\). Substituting these into the formula, we find the standard form of the equation of the parabola to be \(y = (1/100)x^2\)