Problem 22 - College Algebra

Question

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((0,20) ;\) Directrix: \(y=-20\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(y = 5x^2\).

Step 1: Identify the vertex

Since the parabola is symmetric with respect to the x-axis and the focus and directrix are mirror images in relation to the x-axis, the vertex of the parabola is at the origin (0,0).

Step 2: Determine the opening direction

Since the focus (0,20) is above the vertex, the parabola opens upwards.

Step 3: Find the value of a

The distance from the vertex to the focus or to the directrix is |4a|. Hence, |4a| = 20, which means a = 5.

Step 4: Formulate the standard equation

For a parabola opening upwards, the standard form is \(y = ax^2\). Thus, replacing \(a\) with 5 yields the standard form of the parabola: \(y = 5x^2\).