x^2-10x-8y+33=0 or 8y = x^2-10x+33x2−10x−8y+33=0or8y=x2−10x+33 or
8y = (x^2-10x+25)-25+33 or 8y = (x-5)^2 +8 8y=(x2−10x+25)−25+33or8y=(x−5)2+8 or
y= 1/8(x-5)^2 +1y=18(x−5)2+1. Comparing with standard equation y= a(x-h)^2+k ; (h,k) y=a(x−h)2+k;(h,k)being vertex , we get here h=5 ,k =1 , a= 1/8 h=5,k=1,a=18. So vertex is at (5,1)(5,1). The parabola opens upwards since a>0a>0
Vertex is at equidistant from focus and directrix. The distance of vertex from directrix is d = 1/(4|a|) =1/(4*1/8)=2d=14|a|=14⋅18=2 and directrix is below the vertex.
So directrix is y=(1-2)= -1y=(1−2)=−1 . Focus is above the vertex and at 5,(1+2) or (5,3)5,(1+2)or(5,3)
graph{x^2-10x-8y+33=0 [-20, 20, -10, 10]} [Ans]
