# What is the equation in standard form of the parabola with a focus at (42,-31) and a directrix of y= 2?

Sep 7, 2017

$y = - \frac{1}{66} {x}^{2} + \frac{14}{11} x - \frac{907}{22} \leftarrow$ standard form

#### Explanation:

Please observe that the directrix is a horizontal line

$y = 2$

Therefore, the parabola is the type that opens upward or downward; the vertex form of the equation for this type is:

$y = \frac{1}{4 f} {\left(x - h\right)}^{2} + k \text{ [1]}$

Where $\left(h , k\right)$ is the vertex and $f$ is the signed vertical distance from the vertex to the focus.

The x coordinate of the vertex is the same as the x coordinate of the focus:

$h = 42$

Substitute $42$ for $h$ into equation [1]:

$y = \frac{1}{4 f} {\left(x - 42\right)}^{2} + k \text{ [2]}$

The y coordinate of the vertex is halfway between the directrix and the focus:

$k = \frac{{y}_{\text{directrix"+y_"focus}}}{2}$

$k = \frac{2 + \left(- 31\right)}{2}$

$k = - \frac{29}{2}$

Substitute $- \frac{29}{2}$ for $k$ into equation [2]:

$y = \frac{1}{4 f} {\left(x - 42\right)}^{2} - \frac{29}{2} \text{ [3]}$

The equation to find the value of $f$ is:

$f = {y}_{\text{focus}} - k$

$f = - 31 - \left(- \frac{29}{2}\right)$

$f = - \frac{33}{2}$

Substitute $- \frac{33}{2}$ for $f$ into equation [3]:

$y = \frac{1}{4 \left(- \frac{33}{2}\right)} {\left(x - 42\right)}^{2} - \frac{29}{2}$

Simplify the fraction:

$y = - \frac{1}{66} {\left(x - 42\right)}^{2} - \frac{29}{2}$

Expand the square:

$y = - \frac{1}{66} \left({x}^{2} - 84 x + 1764\right) - \frac{29}{2}$

Distribute the fraction:

$y = - \frac{1}{66} {x}^{2} + \frac{14}{11} x - \frac{294}{11} - \frac{29}{2}$

Combine like terms:

$y = - \frac{1}{66} {x}^{2} + \frac{14}{11} x - \frac{907}{22} \leftarrow$ standard form

Sep 9, 2017

$y = - \frac{1}{66} {x}^{2} + \frac{14}{11} x - \frac{907}{22} ,$

#### Explanation:

We will solve this Problem using the following Focus-Directrix

Property (FDP) of the Parabola.

FDP : Any point on a Parabola is equidistant from the

Focus and the Directrix.

Let, the point $F = F \left(42 , - 31\right) , \text{ and, the line } d : y - 2 = 0 ,$ be

the Focus and the Directrix of the Parabola, say S.

Let, $P = P \left(x , y\right) \in S ,$ be any General Point.

Then, using the Distance Formula, we have, the distance,

$F P = \sqrt{{\left(x - 42\right)}^{2} + {\left(y + 31\right)}^{2}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(1\right) .$

Knowing that the $\bot -$dist. between a point $\left(k , k\right) ,$ and, a line :

$a x + b y + c = 0 ,$ is, $| a h + b k + c \frac{|}{\sqrt{{a}^{2} + {b}^{2}}} ,$ we find that,

$\text{the "bot-"dist. btwn "P(x,y), &, d" is, } | y - 2 | \ldots \ldots \ldots \ldots . . \left(2\right) .$

By FDP, $\left(1\right) , \mathmr{and} \left(2\right) ,$ we have,

$\sqrt{{\left(x - 42\right)}^{2} + {\left(y + 31\right)}^{2}} = | y - 2 | , \mathmr{and} ,$

${\left(x - 42\right)}^{2} = {\left(y - 2\right)}^{2} - {\left(y + 31\right)}^{2} = - 66 y - 957 , i . e . ,$

${x}^{2} - 84 x + 1764 = - 66 y - 957.$

$\therefore 66 y = - {x}^{2} + 84 x - 2721 ,$ which, in the Standard Form,

reads, $y = - \frac{1}{66} {x}^{2} + \frac{14}{11} x - \frac{907}{22} ,$

as Respected Douglas K. Sir has already derived!

Enjoy Maths.!