Chapter 8.1, Problem 28E

Solution

To calculate : The equation in the standard form for the parabola that satisfies conditions vertex $\left(-3,-3\right)$ opens downward and focal width as $20$ .

The equation in the standard form for the parabola that satisfies conditions vertex $\left(-3,-3\right)$ opens downward and focal width as $20$ is ${\left(x+3\right)}^2=-20\left(y-3\right)$ .

Given information :

The parabola with vertex $\left(-3,-3\right)$ opens downward and focal width as $20$ .

Formula used :

The equation for parabola for which vertex is at $\left(h,k\right)$ and opens downward is ${\left(x-h\right)}^2=4p\left(y-k\right)$ .

Calculation :

Consider the parabola with vertex $\left(-3,-3\right)$ opens downward and focal width as $20$ ,

As parabola opens to the downward, then $p<0$ .

The width of focal will be $\left|4p\right|$ .

Equate $\left|4p\right|$ to $20$ .

  $\begin{array}{c}\left|4p\right|=20\\ \left|p\right|=5\end{array}$

This gives $p$ as either $5$ or $-5$ .

As $p<0$ , the value of $p$ is $-5$ .

Substitute the desired values to the equation ${\left(x-h\right)}^2=4p\left(y-k\right)$ .

  $\begin{array}{c}{\left(x-\left(-3\right)\right)}^2=4\left(-5\right)\left(y-3\right)\\ {\left(x+3\right)}^2=-20\left(y-3\right)\end{array}$

Thus, the equation in the standard form for the parabola that satisfies conditions vertex $\left(-3,-3\right)$ opens downward and focal width as $20$ is ${\left(x+3\right)}^2=-20\left(y-3\right)$ .