What is the standard form equation of a parabola with a vertex of (5, 6) and a directrix at y= 23/4?

Question

What is the standard form equation of a parabola with a vertex of (5, 6) and a directrix at y= 23/4?

 

standard form equation of a parabola

Answer 1

The most straightforward approach to problems like this to determine the p value.   The distance between the directrix and the vertex is equal to p ( the p value).   For this problem   p =  6 - 23/4  =  1/4.  

   (The distance from the vertex to the focus is also equal to p)

 

   The vertex form of a parabola is   y =  a ( x - x_v)²  +  y_v   where (x_v , y_v) is the location of the vertex. 

For this problem,   x_v = 5  and y_v = 6.    

  The remaining general  fact about parabolas  that is needed is that   a = 1/(4 p)   

 So for this problem,   a = 1  .  Plugging in for a, x_v , y_v  the vertex form becomes

  y = (x -5)² + 6.       Expanding this gives the standard form

  y = x² - 10 x + 31