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

##### 2 Answers

Standard form is:

#### Explanation:

Because the directrix is a vertical line,

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:

Substitute -7 for k into equation [1]:

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:

Substitute 8 for h into equation [2]:

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

Substitute 3 for f into equation [3]:

We will multiply the denominator and write -- as +

Expand the square:

Distribute the

Combine the constant terms:

#x=y^2/12+7/6y+145/12#

#### Explanation:

Directrix

Focus

From this we can findout the vertex.

Look at the diagram

Vertex lies exactly in between Directrix and Focus

#x,y=(5+11)/2, (-7 + (-7))/2=(8, -7)#

The distance between Focus and vertex is

The parabola is opening to the right

The equation of the Parabola here is -

#(y-k)^2=4a(x-h)#

#(h,k)# is the vertex

#h=8#

#k=-7#

Plugin

#(y-(-7))^2=4.3(x-8)#

#(y+7)^2=4.3(x-8)#

#12x-96=y^2+14y+49# [by transpose]

#12x=y^2+14y+49+96#

#12x=y^2+14y+145#

#x=y^2/12+14/12y+145/12#

#x=y^2/12+7/6y+145/12#