Question

1. $x = \frac{1}{4}(y - 5)^2+2$

vertex: (2, 5)
c = 2
focus = (4, 5)
directrix: x = - 5

Explanation:

Step1: Identify vertex form

The equation of a parabola $x = a(y - k)^2+h$ has vertex $(h,k)$. For $x=\frac{1}{4}(y - 5)^2+2$, the vertex is $(2,5)$.

Step2: Determine the value of $c$

For a parabola of the form $x=a(y - k)^2+h$, the relationship between $a$ and $c$ is $a=\frac{1}{4c}$. Given $a = \frac{1}{4}$, then $\frac{1}{4}=\frac{1}{4c}$, so $c = 1$.

Step3: Find the focus

The focus of a parabola $x=a(y - k)^2+h$ is $(h + c,k)$. Substituting $h = 2,k = 5,c = 1$, we get the focus $(2 + 1,5)=(3,5)$.

Step4: Find the directrix

The directrix of a parabola $x=a(y - k)^2+h$ is $x=h - c$. Substituting $h = 2,c = 1$, we get $x=2-1=1$.

Answer:

Vertex: $(2,5)$
$c = 1$
Focus: $(3,5)$
Directrix: $x = 1$