Question

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

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 vertex is at \((0,0)\) and the focus is at \((0,p)\). If \(p>0\), it opens up; if \(p < 0\), it opens down.

Step2: Compare the given equation with the standard form

The given equation is \(x^{2}=- 20y\). Comparing it with \(x^{2}=4py\), we have \(4p=-20\).

Step3: Solve for \(p\)

Divide both sides of \(4p = - 20\) by 4: \(p=\frac{-20}{4}=-5\).

Step4: Find the focus

Since the standard form for a parabola opening up/down is \(x^{2}=4py\) with focus \((0,p)\), and we found \(p = - 5\), the focus is at \((0,-5)\).

Answer:

D. \((0, - 5)\)