Find the Parabola with Focus (3,-1) and Directrix y=1 (3,-1) y=1

$(3, - 1)$ $y = 1$

Step 1

Since the directrix is vertical, use the equation of a parabola that opens up or down.

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

Step 2

Find the vertex.

The vertex $(h, k)$ is halfway between the directrix and focus. Find the $y$ coordinate of the vertex using the formula $y = \frac{y coordinate of focus + directrix}{2}$ . The $x$ coordinate will be the same as the $x$ coordinate of the focus.

$(3, \frac{- 1 + 1}{2})$

Simplify the vertex.

Add $- 1$ and $1$ .

$(3, \frac{0}{2})$

Divide $0$ by $2$ .

$(3, 0)$

Step 3

Find the distance from the focus to the vertex.

The distance from the focus to the vertex and from the vertex to the directrix is $| p |$ . Subtract the $y$ coordinate of the vertex from the $y$ coordinate of the focus to find $p$ .

$p = - 1 - 0$

Subtract $0$ from $- 1$ .

$p = - 1$

Step 4

Substitute in the known values for the variables into the equation ${(x - h)}^{2} = 4 p (y - k)$ .

${(x - 3)}^{2} = 4 (- 1) (y - 0)$

Step 5

Simplify.

${(x - 3)}^{2} = - 4 y$

Step 6

Find the Parabola with Focus (3,-1) and Directrix y=1 (3,-1) y=1

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