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).

Focus: $F\left(-\frac{1}{12},0\right)$

Short answer

The equation of the parabola with vertex at origin and focus $F\left(-\frac{1}{12},0\right)$ is $y^2=-\frac{1}{3}x$.

Step-by-step solution

Step 1. Given information.

The vertex of the parabola is $\text{V(0,0)}$ and the focus is $F\left(-\frac{1}{12},0\right)$ .

Step 2. Find the equation of the parabola.

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

Vertex of the parabola is$\text{V(0,0)}$
Focus of the parabola is$F\left(-\frac{1}{12},0\right)$
The standard form of the focus is$F(p,0)$.
We see that,$p=-\frac{1}{12}$.

Directrix:

$\begin{array}{l}x=-p\\ x=\frac{1}{12}\end{array}$

The standard form of the parabola is,

$x^2=4py$

Substituting the value of$x=\frac{1}{12}$in the standard form,

$\begin{array}{l}\Rightarrow y^2=4\times \left(\frac{-1}{12}\right)\times x\\ \Rightarrow y^2=\frac{-1}{3}x\end{array}$

Hence, the equation of the parabola is, $y^2=-\frac{1}{3}x$ .