Question

1. find missing attributes & equation. focus (1,1), and directrix is y = 5

vertex: (6,5)
c = ________
y = ________

Explanation:

Step1: Find the vertex

The vertex of a parabola is the mid - point between the focus and the directrix. The x - coordinate of the vertex is the same as the x - coordinate of the focus, which is $x = 1$. The y - coordinate of the vertex is the mid - point between the y - coordinate of the focus ($y_1=1$) and the y - value of the directrix ($y = 5$). So, $y=\frac{1 + 5}{2}=3$. The vertex is $(1,3)$.

Step2: Calculate the value of $c$

The value of $c$ is the distance between the vertex and the focus (or the vertex and the directrix). Since the vertex is $(1,3)$ and the focus is $(1,1)$, $c=\vert3 - 1\vert = 2$. The parabola opens downwards because the focus $(1,1)$ is below the directrix $y = 5$.

Step3: Write the equation of the parabola

The standard form of a parabola that opens up or down is $(x - h)^2=4c(y - k)$, where $(h,k)$ is the vertex. Here, $h = 1,k = 3,c=- 2$ (negative because it opens downwards). So the equation is $(x - 1)^2=-8(y - 3)$.

Answer:

Vertex: $(1,3)$
$c = 2$
$y=-\frac{1}{8}(x - 1)^2+3$