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: $F\left(0,-\frac{3}{4}\right)$

Short answer

The equation of the parabola with vertex at origin and focus $F\left(0,-\frac{3}{4}\right)$ is $x^2=-3y$.

Step-by-step solution

Step 1. Given information.

The vertex of the parabola is$\text{V(0,0)}$ and the focus is $F\left(0,-\frac{3}{4}\right)$ .

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$F\left(0,-\frac{3}{4}\right)$
The standard form of the focus is$F(0,p)$.
We see that,$p=-\frac{3}{4}$.

Directrix:

$\begin{array}{l}y=-p\\ y=\frac{3}{4}\end{array}$

The standard form of the parabola is,

$x^2=4py$

Substituting the value of$y=\frac{3}{4}$in the standard form,

$\begin{array}{l}\Rightarrow x^2=4\times \frac{-3}{4}\times y\\ \Rightarrow x^2=-3y\end{array}$

Hence, the equation of the parabola is, $x^2=-3y$ .