# What is the vertex form of the equation of the parabola with a focus at (55,45) and a directrix of y=47 ?

Mar 31, 2018

$y = - \frac{1}{4} {\left(x - 55\right)}^{2} + 46$

#### Explanation:

Parabola is locus of a point which moves so that its distance from a given point calld focus and a given line ccalled directrix is always constant.

Let the point be $\left(x , y\right)$. Here focus is $\left(55 , 45\right)$ and distance from focus is $\sqrt{{\left(x - 55\right)}^{2} + {\left(y - 45\right)}^{2}}$. Its distance from directrix $y = 47$ i.e. $y - 47 = 0$ is $| y - 47 |$.

Hence equaion of parabola is

(x-55)^2+(y-45)^2)=|y-47|^2

or ${\left(x - 55\right)}^{2} + {y}^{2} - 90 y + 2025 = {y}^{2} - 94 y + 2209$

or ${\left(x - 55\right)}^{2} - 184 = - 4 y$

i.e. $y = - \frac{1}{4} {\left(x - 55\right)}^{2} + 46$

Hence parabola's equation is $y = - \frac{1}{4} {\left(x - 55\right)}^{2} + 46$ and vertex is $\left(55 , 46\right)$.

graph{((x-55)^2-184+4y)(y-47)((x-55)^2+(y-45)^2-0.1)=0 [35, 74.99, 35.02, 55.02]}