# What is the vertex form of the parabola with a focus at (3,5) and vertex at (1,3)?

Jun 6, 2017

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

#### Explanation:

Vertex form of a parabola can be expressed as

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

or

$4 p \left(y - k\right) = {\left(x - h\right)}^{2}$

Where $4 p = \frac{1}{a}$ is the distance between the vertex and the focus.

The distance formula is

$\frac{1}{a} = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

Let's call $\left({x}_{1} , {y}_{1}\right) = \left(3 , 5\right)$ and $\left({x}_{2} , {y}_{2}\right) = \left(1 , 3\right)$. So,

$\frac{1}{a} = \sqrt{{\left(1 - 3\right)}^{2} + {\left(3 - 5\right)}^{2}} = \sqrt{{\left(- 2\right)}^{2} + {\left(- 2\right)}^{2}} = 2 \sqrt{2}$

Cross multiplying gives $a = \frac{1}{2 \sqrt{2}} = \frac{\sqrt{2}}{4}$

The final, vertex form is therefore,

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