Question

the shape of a satellite dish can be described as parabolic. satellite dishes are this shape because radio waves are reflected from the surface of the dish and received into the focus. if the graph of the satellite dish is given by the equation $x^2 = 8y$, what are the coordinates of the focus? (\boxed{\quad},\boxed{\quad})

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 focus is at \((0, p)\).

Step2: Compare the given equation with the standard form

The given equation is \(x^{2}=8y\). Comparing it with \(x^{2}=4py\), we can set \(4p = 8\).

Step3: Solve for \(p\)

To find \(p\), we solve the equation \(4p=8\). Dividing both sides by 4, we get \(p = \frac{8}{4}=2\).

Step4: Determine the focus coordinates

Since the standard form \(x^{2}=4py\) has its focus at \((0, p)\) and we found \(p = 2\), the focus of the parabola \(x^{2}=8y\) is at \((0, 2)\).

Answer:

\((0, 2)\)