Problem 70 - College Algebra

Question

Explain how to use \(y^{2}=8 x\) to find the parabola's focus and directrix.

Step-by-Step Solution

Answer

The focus of the parabola is located at (2, 0) and its directrix is \(x = -2\).

Step 1: Identify the Vertex

In our case, the equation is rewritten as \((y-0)^2 = 2(4)(x-0)\). Hence, by comparing this to the standard form, it can be seen that the vertex h = 0 and k = 0, so the vertex is (0,0).

Step 2: Identify the value of 'a'

Once the vertex is determined, next we need the value of 'a' to find the focus and directrix. In this case, 4a = 8 , so 'a' = 2.

Step 3: Find the Focus

The focus of a parabola that opens to the right is given by \((h+a, k)\). Substituting h = 0 and a = 2, the Focus = (2, 0).

Step 4: Determine the Directrix

The equation for a vertical directrix to the right opening parabola is \(x = h - a\). Substituting h = 0 and a = 2, we get the directrix as \(x = -2\).