# What is the equation of a parabola with a vertex at (3,4) and a focus at (6,4)?

Nov 29, 2015

In vertex form:

$x = \frac{1}{12} {\left(y - 4\right)}^{2} + 3$

#### Explanation:

Since the vertex and focus lie on the same horizontal line $y = 4$, and the vertex is at $\left(3 , 4\right)$ this parabola can be written in vertex form as:

$x = a {\left(y - 4\right)}^{2} + 3$

for some $a$.

This will have its focus at $\left(3 + \frac{1}{4 a} , 4\right)$

We are given that the focus is at $\left(6 , 4\right)$, so:

$3 + \frac{1}{4 a} = 6$.

Subtract $3$ from both sides to get:

$\frac{1}{4 a} = 3$

Multiply both sides by $a$ to get:

$\frac{1}{4} = 3 a$

Divide both sides by $3$ to get:

$\frac{1}{12} = a$

So the equation of the parabola may be written in vertex form as:

$x = \frac{1}{12} {\left(y - 4\right)}^{2} + 3$