Find the Parabola with Focus (1,-4) and Directrix x=2 (1,-4) x=2

$(1, - 4)$ $x = 2$

Step 1

Since the directrix is horizontal, use the equation of a parabola that opens left or right.

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

Step 2

Find the vertex.

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

$(\frac{1 + 2}{2}, - 4)$

Add $1$ and $2$ .

$(\frac{3}{2}, - 4)$

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 $x$ coordinate of the vertex from the $x$ coordinate of the focus to find $p$ .

$p = 1 - \frac{3}{2}$

Simplify.

Write $1$ as a fraction with a common denominator.

$p = \frac{2}{2} - \frac{3}{2}$

Combine the numerators over the common denominator.

$p = \frac{2 - 3}{2}$

Subtract $3$ from $2$ .

$p = \frac{- 1}{2}$

Move the negative in front of the fraction.

$p = - \frac{1}{2}$

Step 4

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

${(y + 4)}^{2} = 4 (- \frac{1}{2}) (x - \frac{3}{2})$

Step 5

Simplify.

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

Step 6

Find the Parabola with Focus (1,-4) and Directrix x=2 (1,-4) x=2

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