Question

Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s).

Directrix: $x=\frac{-1}{8}$

Short answer

The equation of the parabola with vertex at the origin and the directrix $x=\frac{-1}{8}$ is $y^2=\frac{1}{2}x$.

Step-by-step solution

Step 1. Given information.

Given: The vertex of the parabola is $\text{V(0,0)}$ and the directrix is $x=\frac{-1}{8}$ .

Step 2. Find the equation of the parabola

Substituting the value of$\text{p}$in the standard form of the parabola.

The vertex of the parabola is$\text{V(0,0)}$
The directrix of the parabola is$x=\frac{-1}{8}$
The standard form of the directrix is$\text{y=-p}$
We see that,

$\begin{array}{l}-p=-\frac{1}{8}\\ \Rightarrow p=\frac{1}{8}\end{array}$

Focus is

$\begin{array}{l}\text{F(p,0)}\\ \text{F(}\frac{-1}{10}\text{,0)}\end{array}$

The standard form of the parabola is,

$x^2=4py$

Substituting the value of$\text{p}$in the standard form,

$\begin{array}{l}\Rightarrow y^2=4\times \frac{1}{8}\times x\\ \Rightarrow y^2=\frac{1}{2}x\end{array}$

Hence, the equation of the parabola is $y^2=\frac{1}{2}x$ .