Problem 58 - 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 \(y^2 = 8x\) is at the point (2, 0) and the directrix is at \(x = -2\).

Step 1: Identify the value of 'a'

The equation of the parabola is given as \(y^2 = 8x\). This equation is in the form of \(y^2 = 4ax\). Comparing the two, it is clear that 4a = 8. We can solve for 'a' by dividing 8 by 4. Therefore, \(a = 2\).

Step 2: Find the Focus

The focus of the parabola in this form is at the point \((a, 0)\). We have calculated 'a' as 2, so the focus is at the point \((2, 0)\).

Step 3: Find the Directrix

The equation of the directrix for a parabola in this form is \(x = -a\). With 'a' calculated as 2, this gives the equation of the directrix as \(x = -2\).