Question

a parabola is represented by the equation $x^2 = 4y$. what are the coordinates of the focus of the parabola? $\bigcirc\\ (0,4)$ $\bigcirc\\ (0,1)$ $\bigcirc\\ (1,0)$ $\bigcirc\\ (4,0)$

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

Step2: Compare the given equation with the standard form

The given equation is \(x^{2}=4y\). Comparing it with \(x^{2}=4py\), we can see that \(4p = 4\). Solving for \(p\), we divide both sides by 4: \(p=\frac{4}{4}=1\).

Step3: Determine the focus coordinates

Since the focus of the parabola \(x^{2}=4py\) is at \((0,p)\) and \(p = 1\), the focus is at \((0,1)\).

Answer:

\((0,1)\) (corresponding to the option: (0,1))