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Find the standard form of the equation of the parabola with a focus at (5, 0) and a directrix at x = -5.

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4 Answers

Let me add some more details abut the parabola before applying a formula, 
 
1. Recognize the type, vertical(regular) or horizontal(sideways).
    The difference in the two types is in which variable is squared:
    for vertical parabolas, the x part is squared  and for horizontal, the y part is squared.
Here, for the question, the directrix is x=-5, it means that the graph type is horizontal.
 
2. Remember or you can derive the formula from y= ax2 + bx +c for the vertical, or x= ay2 + by + c for the horizontal,
such that, the vertex form of a parabola equation with its vertex at (h, k) as, 
  y = a(x – h)2 + k  → 1/a( y - k) = (x - h)2,   for the vertical
  x = a(y – k)2 + h, →  1/a(x - h )= (y - k)2,   for the horizontal
 
Or, the conics form of the parabola equation as

  4p(y – k) = (x – h)2, for the vertical 
  4p(x – h) = (y – k)2, for the horizontal
 
where 4p = 1/a, or a= 1/(4p).  
Keep h with x, y with k, and p with the first degree part (not squared part)
 
The conic form is more useful because the variable p represents the distance between the vertex and the focus.  Also, the same p represents the distance from the vertex to the directrix.  The vertex is exactly midway between the directrix and the focus.  Obviously, 2p represents the distance between the focus and the directrix.
 
3. Find the variables for the formula recognized.
For the given question, the formula to be 
    4p(x – h) = (y – k)2, for the horizontal
 
From the focus (5,0) and the directrix is x=-5 given,
the value of h (which stays with x) is located in the middle of 5 and -5, that is 0,
and k (which stays with y) is the same y value of the focus, that is 0, therefore the vertex is at (0,0)
P is the distance of x-axix between the vertex and the focus, that is p = 5-0 = 5
Replace all the variables into the formula:  4(5)(x - 0) = (y - 0)2  →  20x = y2,  or y2 = 20x
 
I hope that this explanation helps to understand the parabola, and to remember its formula.  This is one of the formula easy to get confused and forget.
 
Kate,
 
First of all if we sketch what we're given on a graph, we see that the directrix is a vertical line @ x = -5, and the focus @ (5,0) lies on the x-axis.  This tells us that our parabola is going to open to the right (+ x direction).  Next, the vertex has to lie half-way between the focus and the directrix, which means it will have coordinates x = 0, y =0 so the vertex is the point (0,0).
 
For a parabola opening to the right, the standard equation is given by:
 
(x-h)² = +4p(y-k)
 
h = x coordinate of the vertex = 0 in our case, k = y coordinate of the vertex also = 0.
 
p is the distance between the vertex and the focus = 5
 
So our equation is then (x-0)² = 4(5)(y-0)
 
Simplifying x² = 20y
You need to memorize the standard form or find it in a textbook.
 
For a parabola with a horizontal axis such as this example, the equation is
 
(y - k)2 = 4p(x - h)
 
(h , k) is the vertex
 
directrix equation is x = h - p
 
Hint: the vertex lies half way between the focus point and the directrix