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: $F(0,6)$

Short answer

The equation of the parabola with vertex at origin and focus$\text{F(0,6)}$ is $x^2=24y$.

Step-by-step solution

Step 1.Given information.

The vertex of the parabola is $\text{V(0,0)}$ and the focus is $\text{F(0,6)}$ .

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$\text{F(0,6)}$
The standard form of the focus is$\text{F(0,p)}$.
We see that,$\text{p=6}$.

Directrix:

$\begin{array}{l}y=-p\\ y=-6.\end{array}$

The standard form of the parabola is,

$x^2=4py$

Substituting the value of$\text{p=6}$in the standard form,

$\begin{array}{l}\Rightarrow x^2=4\times 6\times y\\ \Rightarrow x^2=24y.\end{array}$

Hence, the equation of the parabola is, $x^2=24y$ .