# What is the equation of the parabola with a focus at (44,55) and a directrix of y= 66?

Oct 21, 2017

${x}^{2} - 88 x + 22 y + 605 = 0$

#### Explanation:

Parabola is the locus of a point which moves so that its distances from a given point called focus and from a given line called directrix are equal.

Here let us consider the point as $\left(x , y\right)$. Its distance from focus $\left(44 , 55\right)$ is $\sqrt{{\left(x - 44\right)}^{2} + {\left(y - 55\right)}^{2}}$

and as distance of a point x_1,y_1) from a line $a x + b y + c = 0$ is $| \frac{a {x}_{1} + b {y}_{1} + c}{\sqrt{{a}^{2} + {b}^{2}}} |$, distance of $\left(x , y\right)$ from $y = 66$ or $y - 66 = 0$ (i.e. $a = 0$ and $b = 1$) is $| y - 66 |$.

Hence equation of parabola is

${\left(x - 44\right)}^{2} + {\left(y - 55\right)}^{2} = {\left(y - 66\right)}^{2}$

or ${x}^{2} - 88 x + 1936 + {y}^{2} - 110 y + 3025 = {y}^{2} - 132 y + 4356$

or ${x}^{2} - 88 x + 22 y + 605 = 0$

The parabola along with focus and directrix appears as shown below.

graph{(x^2-88x+22y+605)((x-44)^2+(y-55)^2-6)(y-66)=0 [-118, 202, -82.6, 77.4]}

Oct 21, 2017

$y = - \frac{1}{18} \left({x}^{2} - 88 x + 847\right)$

#### Explanation:

Focus $\left(44 , 55\right)$
Directrix $y = 66$

Vertex $\left(44 , \frac{55 + 66}{2}\right) = \left(44 , 60.5\right)$

Distance between vertex and focus $a = 60.5 - 55 = 4.5$

Since Directrix is above vertex, this parabola opens down.
Its equation is -

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

Where -

$h = 44$
$k = 60.5$
$a = 4.5$

${\left(x - 44\right)}^{2} = - 4 \times 4.5 \left(y - 60.5\right)$
${x}^{2} - 88 x + 1936 = - 18 y + 1089$

$- 18 y + 1089 = {x}^{2} - 88 x + 1936$

$- 18 y = {x}^{2} - 88 x + 1936 - 1089$
$- 18 y = {x}^{2} - 88 x + 847$

$y = - \frac{1}{18} \left({x}^{2} - 88 x + 847\right)$