# What is the equation of a parable with vertex (2,2) and the directrix is y=2.5?

Dec 7, 2016

The equation is $2 y - 4 = - {\left(x - 2\right)}^{2}$

#### Explanation:

If the directrix is $y = 2.5$ and the vertex is $\left(2 , 2\right)$

The line of symmetry is $x = 2$

and the focus is $= \left(2 , 1.5\right)$

The distance of a point $\left(x , y\right)$ on the parabola to the directrix is equal to the distance of that point to the focus.

$2.5 - y = \sqrt{{\left(x - 2\right)}^{2} + {\left(y - 1.5\right)}^{2}}$

${\left(2.5 - y\right)}^{2} = {\left(x - 2\right)}^{2} + {\left(y - 1.5\right)}^{2}$

$6.25 - 5 y + {y}^{2} = {\left(x - 2\right)}^{2} + {y}^{2} - 3 y + 2.25$

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

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

graph{(2y-4+(x-2)^2)(y-2.5)=0 [-11.25, 11.25, -5.63, 5.62]}