Chapter 9.1, Problem 74E

To determine

To find: The vertex, focus and directrix of the parabola $x^2-2x+8y+9=0$ and verify with the help of graphing utility.

Answer to Problem 74E

The vertex, focus and directrix of the parabola $x^2-2x+8y+9=0$ are $\left(1,-1\right)$ , $\left(1,-3\right)$ , $y=1$ respectively. The graph of the parabola with vertex, focus and directrix is shown in Figure (1).

Explanation of Solution

Given information:

The equation of the parabola is $x^2-2x+8y+9=0$ .

Calculation:

Simplify the equation of parabola.

  $\begin{array}{c}{x}^2-2x+8y+9=0\\ x^2-2x+1^2-1^2+8y+9=0\\ {\left(x-1\right)}^2+8y+9-1=0\\ {\left(x-1\right)}^2=-8y-8\\ {\left(x-1\right)}^2=-8\left(y+1\right)\end{array}$

Compare the given equation of parabola with general vertical axis parabola equation ${\left(x-h\right)}^2=4p\left(y-k\right)$ . This gives the value of $h$ , $k$ , $p$ as $1$ , $-1$ , $-2$ respectively.

The vertex of the parabola $x^2-2x+8y+9=0$ is $\left(h,k\right)=\left(1,-1\right)$ .

The focus of the horizontal axis parabola $x^2-2x+8y+9=0$ is:

  $\begin{array}{c}\left(h,k+p\right)=\left(1,-1+\left(-2\right)\right)\\ =\left(1,-3\right)\end{array}$

The directrix of the parabola $x^2-2x+8y+9=0$ is:

  $\begin{array}{c}y=k-p\\ =-1-\left(-2\right)\\ =1\end{array}$

Therefore, the vertex, focus and directrix of the parabola $x^2-2x+8y+9=0$ are $\left(1,-1\right)$ , $\left(1,-3\right)$ , $y=1$ respectively.

The graph of the parabola $x^2-2x+8y+9=0$ with vertex, focus and directrix is shown below.

  

Figure (1)

Here we can verify the vertex, focus, directrix of the parabola $x^2-2x+8y+9=0$ are $\left(1,-1\right)$ , $\left(1,-3\right)$ , $y=1$ respectively.