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: $y=\frac{1}{10}$

Short answer

The equation of the parabola with vertex at the origin and the directrix $y=\frac{1}{10}$ is $x^2=\frac{-2}{5}y$.

Step-by-step solution

Step 1. Given information.

Given: The vertex of the parabola is $\text{V(0,0)}$ and the directrix is $y=\frac{1}{10}$ .

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$y=\frac{1}{10}$
The standard form of the directrix is$\text{y=-p}$
We see that

$\begin{array}{l}-p=\frac{1}{10}\\ \Rightarrow p=\frac{-1}{10}\end{array}$

Focus is

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

The standard form of the parabola is,

$x^2=4py$

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

$\begin{array}{l}\Rightarrow x^2=4\times \left(\frac{-1}{10}\right)\times y\\ \Rightarrow x^2=\frac{-2}{5}y\end{array}$

Hence, the equation of the parabola is $x^2=\frac{-2}{5}y$ .