Chapter 10.6, Problem 3CFU

To determine

The rectangular and parametric equation of a parabola with vortex at the origin and opening at the left.

Explanation of Solution

The standard form of equation of a parabola is,

  $\begin{array}{l}$ {(y-k)}^2=4P{(x-h)}^2$\text{where,}\\ h,k\text{ represents the coordinates of the vortex}\text{.}\\ \text{Now, if }P\text{ is greater than zero then the parabola opens in the left}\text{. }\end{array}$

Here it is given that the vortex of the parabola is at the origin.

Therefore the equation of the parabola is,

  $\begin{array}{l}$ {(y-k)}^2=4P{(x-h)}^2$$ {(y-0)}^2=4P{(x-0)}^2$y^2=4P{x}^2\end{array}$

Now, let us take p=1,

Therefore the equation becomes,

  $y^2=4x^2$

Now we convert the rectangular equation into parametric equation,

  $\begin{array}{l}{y}^2=4x^2\\ let,\\ x=t\\ therefore,\\ y^2=4t^2\\ y=\sqrt{4t^2}=2t\end{array}$

Therefore the parametric equation is,

  $\begin{array}{l}x=t\\ \&,\\ y=2t\end{array}$