# What is the standard form equation of the parabola with a directrix of x=5 and focus at (11, -7)?

May 3, 2017

Standard form is:

$x = \frac{1}{12} {y}^{2} + \frac{14}{12} y + \frac{145}{12}$

#### Explanation:

Because the directrix is a vertical line, $x = 5$, the vertex form for the equation of the parabola is:

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

where (h,k) is the vertex and f is the signed horizontal distance from the vertex to the focus.

We know that the y coordinate, k, of the vertex is the same as the y coordinate of the focus:

$k = - 7$

Substitute -7 for k into equation [1]:

$x = \frac{1}{4 f} {\left(y - - 7\right)}^{2} + h \text{ [2]}$

We know that the x coordinate of the vertex is the midpoint between the x coordinate of the focus and the x coordinate of the directrix:

$h = \frac{{x}_{\text{focus"+x_"directrix}}}{2}$

$h = \frac{11 + 5}{2}$

$h = \frac{16}{2}$

$h = 8$

Substitute 8 for h into equation [2]:

$x = \frac{1}{4 f} {\left(y - - 7\right)}^{2} + 8 \text{ [3]}$

The focal distance is the signed horizontal distance from the vertex to the focus:

$f = {x}_{\text{focus}} - h$

$f = 11 - 8$

$f = 3$

Substitute 3 for f into equation [3]:

$x = \frac{1}{4 \left(3\right)} {\left(y - - 7\right)}^{2} + 8$

We will multiply the denominator and write -- as +

$x = \frac{1}{12} {\left(y + 7\right)}^{2} + 8$

Expand the square:

$x = \frac{1}{12} \left({y}^{2} + 14 y + 49\right) + 8$

Distribute the $\frac{1}{12}$

$x = \frac{1}{12} {y}^{2} + \frac{14}{12} y + \frac{49}{12} + 8$

Combine the constant terms:

$x = \frac{1}{12} {y}^{2} + \frac{14}{12} y + \frac{145}{12}$

May 3, 2017

$x = {y}^{2} / 12 + \frac{7}{6} y + \frac{145}{12}$

#### Explanation:

Directrix $x = 5$
Focus $\left(11 , - 7\right)$
From this we can findout the vertex.
Look at the diagram

Vertex lies exactly in between Directrix and Focus

$x , y = \frac{5 + 11}{2} , \frac{- 7 + \left(- 7\right)}{2} = \left(8 , - 7\right)$

The distance between Focus and vertex is $a = 3$
The parabola is opening to the right
The equation of the Parabola here is -

${\left(y - k\right)}^{2} = 4 a \left(x - h\right)$
$\left(h , k\right)$ is the vertex
$h = 8$
$k = - 7$

Plugin h=8; k=-7 and a=3# in the equation

${\left(y - \left(- 7\right)\right)}^{2} = 4.3 \left(x - 8\right)$
${\left(y + 7\right)}^{2} = 4.3 \left(x - 8\right)$

$12 x - 96 = {y}^{2} + 14 y + 49$ [by transpose]
$12 x = {y}^{2} + 14 y + 49 + 96$
$12 x = {y}^{2} + 14 y + 145$
$x = {y}^{2} / 12 + \frac{14}{12} y + \frac{145}{12}$
$x = {y}^{2} / 12 + \frac{7}{6} y + \frac{145}{12}$