Question

for the given model of the form $x^{2}=4py$,
(a) determine the value of $p$.
(b) identify the focus of the parabola.
(c) write an equation for the directrix.
express numbers in exact, simplest form.
$x^{2}=12y$

Explanation:

Step1: Compare equations

Compare \(x^{2}=12y\) with \(x^{2} = 4py\).
$$4p=12$$

Step2: Solve for \(p\)

Divide both sides by \(4\).
$$p=\frac{12}{4}=3$$

Step3: Find the focus

For a parabola \(x^{2}=4py\), the focus is \((0,p)\). Substitute \(p = 3\), so the focus is \((0,3)\).

Step4: Find the directrix

For a parabola \(x^{2}=4py\), the directrix is \(y=-p\). Substitute \(p = 3\), so \(y=-3\).

Answer:

(a) \(p = 3\)
(b) Focus: \((0,3)\)
(c) Directrix: \(y=-3\)