Question

a parabola can be represented by the equation $x^{2} = -20y$. what are the coordinates of the focus of the parabola? $\circ$ $(-5,0)$ $\circ$ $(5,0)$ $\circ$ $(0,5)$ $\circ$ $(0,-5)$

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola that opens up or down is \(x^{2}=4py\), where the focus is at \((0,p)\). If \(p>0\), it opens up; if \(p < 0\), it opens down.
Given the equation \(x^{2}=- 20y\), we can compare it with the standard form \(x^{2}=4py\). So, \(4p=-20\).

Step2: Solve for \(p\)

To find \(p\), we solve the equation \(4p=-20\). Dividing both sides by 4, we get \(p=\frac{-20}{4}=- 5\).

Step3: Find the focus

Since the standard form \(x^{2}=4py\) has its focus at \((0,p)\), and we found that \(p = - 5\), the focus of the parabola \(x^{2}=-20y\) is at \((0,-5)\).

Answer:

D. \((0, - 5)\) (assuming the options are labeled as A. \((-5,0)\), B. \((5,0)\), C. \((0,5)\), D. \((0,-5)\))