Chapter 8.1, Problem 21E

Solution

To calculate : The equation in the standard form for the parabola that satisfies conditions focus $\left(-2,-4\right)$ and vertex $\left(-4,-4\right)$ .

The equation in the standard form for the parabola that satisfies conditions focus $\left(-2,-4\right)$ and vertex $\left(-4,-4\right)$ is ${\left(y+4\right)}^2=8\left(x+4\right)$ .

Given information :

The parabola with focus $\left(-2,-4\right)$ and vertex $\left(-4,-4\right)$ .

Formula used :

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

Calculation :

Consider the parabola with focus $\left(-2,-4\right)$ and vertex $\left(-4,-4\right)$ ,

It is given that $\left(h,k\right)=\left(-4,-4\right)$ and focus is at $\left(-2,-4\right)$ .

Equate $h+p$ to $-2$ and substitute $-4$ for $h$ .

  $\begin{array}{c}h+p=-2\\ -4+p=-2\\ p=2\end{array}$

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

  $\begin{array}{c}{\left(y-\left(-4\right)\right)}^2=4\left(2\right)\left(x-\left(-4\right)\right)\\ {\left(y+4\right)}^2=8\left(x+4\right)\end{array}$

Thus, the equation in the standard form for the parabola that satisfies conditions focus $\left(-2,-4\right)$ and vertex $\left(-4,-4\right)$ is ${\left(y+4\right)}^2=8\left(x+4\right)$ .