Problem 19 - College Algebra

Question

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((-5,0) ;\) Directrix: \(x=5\)

Step-by-Step Solution

Answer

The equation of the parabola in standard form is \( x^2=-20y \).

Step 1: Identify and Calculate the Vertex

Given the focus \((-5,0)\) and directrix \(x=5\), note that the vertex should be midway between these two points. The axis of symmetry of the parabola is a vertical line through the vertex, so it has equidistant from the focus and directrix. Therefore, the x-coordinate of the vertex is halfway between -5 and 5, which would give us 0. The y-coordinate of the given focus has already been provided as 0, hence the vertex is at the point (0,0).

Step 2: Determine the value of 'a'

'a' is the distance from the focus to the vertex or the vertex to the directrix. Considering the focus (-5,0) and the vertex (0,0), you can determine that this distance is 5.

Step 3: Write the Equation of the Parabola

Depending on the problem specifics, an equation of the form \((x-h)^2=4a(y-k)\) or \((y-k)^2=4a(x-h)\) can be used based on whether it opens up, down, or sideways. Here, since the directrix is a vertical line, the parabola opens to the left or right. Because the focus is to the left of the directrix, the parabola opens to the left, so use the first form. Substituting h=0, k=0 and a=5 into \( (x-h)^2 = 4a(y-k) \), the standard form of the equation of the parabola is \( x^2=-20y \).