Chapter 11.1, Problem 38E

To determine

To find: the equation of the parabola.

Answer to Problem 38E

The equation of parabola is $x^2=-24y$ .

Explanation of Solution

The equation of parabola is $x^2=-24y$ .

Given information:

Vertex $\left(0,0\right)$

Directrix has y-intercept $6$ .

Calculation:because it is given that directrix has y-intercept, so the parabola should be with vertical axis and the intercept is positive so the graph opens downwards so, $p<0$ .

So the equation of parabola should be in the following form,

  $x^2-4py$

For parabola with verticalaxis, directrix is $y=-p$ .

Because Directrix has y-intercept $6$ , so $y=6$ satisfy the equation of directrix.

  $\begin{array}{l}y=-p\\ 6=-p\text{ [substitute }y\text{ by 6]}\\ p=-6\text{ [mulyiply by -1]}\end{array}$

So the equation of parabola is,

  $\begin{array}{l}{x}^2-4py\\ x^2-4\left(-6\right)y\text{ [substitute p by -6]}\\ x^2+24y\text{ [multiply]}\\ x^2+24y=0\text{ [simplify]}\\ x^2=-24y\end{array}$

Hence, the equation of parabola is $x^2=-24y$ .