Chapter 10, Problem 30RE

To determine

Tofind:the equation of conic using its given description and graph the equation.

Answer to Problem 30RE

The parabolic equation is ${\left(\text{x-3}\right)}^{\text{2}}\text{=-4}\left(\text{y-7}\right)$ .

Explanation of Solution

Given:

The parabola:

Focus at $\left(3,6\right)$ and Directrix the line $\text{y=8}$ .

Concept used:

The standard form is ${\left(\text{x-h}\right)}^{\text{2}}\text{=4p}\left(\text{y-k}\right)$ where the focus is $\left(\text{h,k+p}\right)$ and the directrix is $\text{y=k-p}$ . If the parabola is rotated so that its vertex is $\left(\text{h,k}\right)$ and its axis of symmetry is parallel to the x-axis, it has an equation of ${\left(\text{y-k}\right)}^{\text{2}}\text{=4p}\left(\text{x-h}\right)$ where the focus is $\left(\text{h+p,k}\right)$ and the directrix is $\text{x=h-p}$ .

Calculation:

According to the given:

The parabola:

Focus at $\left(3,6\right)$ and Directrix the line $\text{y=8}$ .

It resembles the equation of parabola:

  ${\left(\text{x-h}\right)}^{\text{2}}\text{=-4a}\left(\text{y-k}\right)$ .

Focus= $\left(\text{h,k-a}\right)\text{=}\left(\text{3,6}\right)$ .

Thus $\text{h=3, k- a=6}$ …… $\left(\text{i}\right)$

Directrix the line $\text{y=8}$ .

Directrix: $\text{y=k+a}\Rightarrow \text{8=k+a}$ ……. $\left(\text{ii}\right)$

From equation $\left(\text{i}\right)$ and $\left(\text{ii}\right)$ :

  $\text{k=7,a=1}$ .

Thus, the equation is ${\left(\text{x-3}\right)}^{\text{2}}\text{=-4}\left(\text{y-7}\right)$ .

The equation can be in shown in graph as:

The graph of parabolic equation ${\left(\text{x-3}\right)}^{\text{2}}\text{=-4}\left(\text{y-7}\right)$ .

Hence, parabolic equation is ${\left(\text{x-3}\right)}^{\text{2}}\text{=-4}\left(\text{y-7}\right)$ .