Chapter 10, Problem 21RE

To determine

Tofind:the equation of conic described and graph the equation.

Answer to Problem 21RE

Theequation is parabolic ${\text{y}}^{\text{2}}\text{=-8x}$ .

Explanation of Solution

Given:

Parabola, Focus at $\left(\text{2,0}\right)$ directrix $\Rightarrow x=2$ .

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:

Parabola, Focus at $\left(\text{2,0}\right)$ directrix $\Rightarrow x=2$ .

It resembles parabola with equation

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

Here the ${\text{a=2,h=0,k=0 and y}}^{\text{2}}\text{=-4×2x}\Rightarrow {\text{y}}^{\text{2}}\text{=-8x}\text{.}$

The graphshows that parabola is form at left side of the graph and it intersect at the origin.

The graph of the parabola ${\text{y}}^{\text{2}}\text{=-8x}$ .

  

Hence, the equation is parabolic ${\text{y}}^{\text{2}}\text{=-8x}$ .