Circles and parabola

Question

find the equation of a parabola with vertex 3,5 and directrix y=7

Answer 1

We know that the vertex is the point (3,5) and the directrix is y=7. Draw this on a sheet of graph (or ordinary) paper. Notice the line y=7 is horizontal and the point (3,5) is below it. Then our parabola must open up/down and since the vertex is below the line it must open up downwards. For such a parabola, the equation is

 

(x-h)^2=4p(y-k)

 

where the vertex is the point (3,5) and p is the distance (with appropriate sign) from the vertex to the directrix. Well, the vertex is distance 2 away from the directrix. So |p|=2. But is p positive or negative? Well, looking at our graph the parabola must open downwards so that p needs to be negative. So p=-2. But then we have everything we need! Why? Well we know that (h,k)=(3,5) so that h=3 and k=5. So we only need to plug things in

 

(x-h)^2 = 4p(y-k)

(x-3)^2 = 4(-2)(y-5)

(x-3)(x-3)= -8(y-5)

x^2-6x+9 = -8y+40

x^2-6x+9-40 = -8y

x^2-6x-31=-8y

y= -1/8 x^2 + 3/4 x + 31/8