Problem 78 - Algebra and Trigonometry

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 at point (2,0) and the equation of the directrix is \(x=-2\).

Step 1: Identify the Form of Equation

The equation provided is \(y^{2}=8x\). This is of the form \(y^{2}=4a x\), where \(4a\) is the coefficient of \(x\). Here, \(4a=8\).

Step 2: Calculate the value of \(a\)

To get the value of \(a\), equate \(4a=8\). Solving for \(a\) gives \(a=2\).

Step 3: Determine the Focus

The focus is at the point \((a,0)\), which is \((2,0)\) when \(a=2\).

Step 4: Determine the Directrix

The directrix is a vertical line to the left of the vertex. Its equation is \(x=-a\), which is \(x=-2\) when \(a=2\).