# Question #46d00

Oct 30, 2017

The distance between any point, $\left(x , y\right)$ on the parabola and the focus $\left(0 , 1\right)$ is:

$d = \sqrt{{\left(x - 0\right)}^{2} + {\left(y - 1\right)}^{2}}$

The distance any point, $\left(x , y\right)$, on the parabola and the line $y = - 1$ is:

$d = \sqrt{{\left(y - \left(- 1\right)\right)}^{2}}$

Because the definition of the parabola requires that these two distances be equal, we can set the right sides of both equations equal:

$\sqrt{{\left(x - 0\right)}^{2} + {\left(y - 1\right)}^{2}} = \sqrt{{\left(y - \left(- 1\right)\right)}^{2}}$

Square both sides of the equation and convert the -- to a +:

${\left(x - 0\right)}^{2} + {\left(y - 1\right)}^{2} = {\left(y + 1\right)}^{2}$

Expand the squares:

${x}^{2} + {y}^{2} - 2 y + 1 = {y}^{2} + 2 y + 1$

Combine like terms:

$4 y = {x}^{2}$

Divide both sides by 4:

$y = \frac{1}{4} {x}^{2}$

The derivation is complete.