Derive the equation of the parabola with a focus at (–2, 4) and a directrix of y = 6. Put the equation in standard form.

Question

Hi, I am struggling with how to solve this problem. I would really appreciate the help, thanks!

Answer 1

Hi Jessi,

 

Since we are given the focus and the directrix of the parabola, we'll first determine the vertex form of its equation and then transform it into the standard form.

 

Let's begin by writing the vertex form of the equation for a parabola and identifying its properties:

    y = a(x - h)² + k

         axis of symmetry (x = h)

         focus (h,k + c)

         vertex (h,k)

         directrix (y = k - c)

         a = 1/4c

         

We can see from the above information that the axis of symmetry passes through both the focus and the vertex.  Since the focus is (-2,4), we know that the x-coordinate of the vertex (h) is -2.  We also know that the parabola opens down, because the directrix

(y = 6) is above the focus (-2,4).  Therefore, the sign of coefficient (a) is negative.

 

To calculate the  y-coordinate of the vertex (k), we can use the equations for distance (c) in terms of the focus and the directrix:

    1.  focus = k + c (4 = k + c)

    2.  directrix = k - c  (6 = k - c)

Adding the equations together to eliminate (c):

      4 = k + c

  + (6 = k - c)

   --------------

      2k = 10

        k = 10/2

        k = 5

 

Next, we can plug in our value for (k) using either equation for (c). We'll use the focus equation in terms of (c):

    focus = k + c

    4 = 5 + c

    c = 4 - 5

    c = -1

 

Now that we know (c), we can calculate (a):

    a = 1/4c

    a = 1/4(-1)

    a = -(1/4) 

 

We now have all our variables determined for the vertex form of the equation for this parabola:

    y = a(x - h)² + k

          a = -(1/4)

          h = -2

          k = 5

    y = -(1/4)(x - (-2))² + 5

    y = -(1/4)(x + 2)² + 5

 

Finally, we'll go from vertex form to standard form by multiplying everything out and combining like terms:

    y = -(1/4)(x + 2)² + 5

    y = -(1/4)(x² + 4x + 4) + 5

    y = -0.25x² - x - 1 + 5

    y = -0.25x² - x + 4 (our answer)

 

Below is the link to our graph of this parabola:

https://dl.dropbox.com/s/p06bms6b0lporen/Graph_of_Parabola_with_Focus_Directrix_Vertex.png?raw=1

 

The parabola is graphed in blue along with its properties:

    1.  directrix (horizontal green line).

    2.  axis of symmetry (vertical green line).

    3.  focus and vertex (red points).

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan (Romans 5:8)