Chapter 8.1, Problem 18E

Solution

To calculate : The equation in the standard form for the parabola that satisfies conditions vertex $\left(0,0\right)$ opens to left and focal width as $12$ .

The equation in the standard form for the parabola that satisfies conditions vertex $\left(0,0\right)$ opens to left and focal width as $12$ is $y^2=-12x$ .

Given information :

The parabola with vertex $\left(0,0\right)$ opens to left and focal width as $12$ .

Formula used :

The equation for parabola for which focus $\left(p,0\right)$ is left to the vertex is $y^2=4px$ .

Calculation :

Consider the parabola with vertex $\left(0,0\right)$ opens to left and focal width as $12$ ,

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

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

Equate $\left|4p\right|$ to $12$ .

  $\begin{array}{c}\left|4p\right|=12\\ \left|p\right|=3\end{array}$

This gives $p$ as either $3$ or

  $-3$ .

As $p<0$ , the value of $p$ is $-3$ .

Substitute this value into the equation $y^2=4px$ .

  $\begin{array}{c}{y}^2=4\left(-3\right)x\\ y^2=-12x\end{array}$

Thus, the equation in the standard form for the parabola that satisfies conditions vertex $\left(0,0\right)$ opens to left and focal width as $12$ is $y^2=-12x$ .