Chapter 8.1, Problem 26E

Solution

To calculate : The equation in the standard form for the parabola that satisfies conditions vertex $\left(3,5\right)$ and directrix $y=7$ .

The equation in the standard form for the parabola that satisfies conditions vertex $\left(3,5\right)$ and directrix $y=7$ is ${\left(x-3\right)}^2=-8\left(y-5\right)$ .

Given information :

The vertex $\left(3,5\right)$ and directrix $y=7$ .

Formula used :

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

Calculation :

Consider the vertex $\left(3,5\right)$ and directrix $y=7$ ,

The directrix is horizontal and situated above the vertex, this implies that the parabola opens downward.

Equate $k-p$ to $7$ and substitute $5$ for $k$ .

  $\begin{array}{c}k-p=7\\ 5-p=7\\ p=-2\end{array}$

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

  $\begin{array}{c}{\left(x-3\right)}^2=4\left(-2\right)\left(y-5\right)\\ {\left(x-3\right)}^2=-8\left(y-5\right)\end{array}$

Thus, the equation in the standard form for the parabola that satisfies conditions vertex $\left(3,5\right)$ and directrix $y=7$ is ${\left(x-3\right)}^2=-8\left(y-5\right)$ .