Answer
(a)
Vertex: $V(-1,2)$
Focus: $F(-1,7)$
Directrix: $y=-3$
Work Step by Step
Equation of a parabola with vertical axis and vertex at $(h,k)$:
$(x-h)^2=4p(y-k)$
$x^2+2x-20y+41=0$
$x^2+2x+1=20y-41+1=20y-40$
$(x+1)^2=20(y-2)$
$[x-(-1)]^2=20(y-2)$
$h=-1$
$k=2$
Vertex: $V(-1,2)$
$4p=20$
$p=5$. Parabola opens upward.
The given equation can be obtained by shifting
$x^2=20y$
left 1 unit and upward 2 units. In this equation:
Focus: $F(0,p)=F(0,5)$
Directrix: $y=-p=-5$
Now, shift the vertex and the directrix 1 unit to the left and 2 unit upward:
Focus: $F(-1,7)$
Directrix: $y=-3$
