Question

sections: parabolas
which is the directrix of a parabola with equation $x^2 = 8y$?
$x = -2$
$y = 2$
$x = 2$
$y = -2$

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola that opens upward or downward is \(x^{2}=4py\), where the vertex is at \((0,0)\), the focus is at \((0,p)\), and the directrix is the line \(y = -p\).

Step2: Identify \(p\) from the given equation

Given the equation \(x^{2}=8y\), we compare it with \(x^{2}=4py\). So, \(4p = 8\). Solving for \(p\), we divide both sides by 4: \(p=\frac{8}{4}=2\).

Step3: Find the directrix

Since the directrix for the parabola \(x^{2}=4py\) is \(y=-p\), and we found \(p = 2\), then the directrix is \(y=-2\).

Answer:

\(y = -2\) (the option with \(y=-2\))