Question

Find the coordinates of the focus and equation of the directrix for the parabola given by y2 = −4x. The general formula for this parabola is y2 = 4px. Therefore, the value of p is . The coordinates of the focus are . The equation of the directrix is .

Answer 1

The general formula for this parabola is y2 = 4px.

Therefore, the value of p is

✔ –1

.

The coordinates of the focus are

✔ (–1,0)

.

The equation of the directrix is

✔ x = 1

.

Explanation:

Answer 2

The coordinates of the focus are
`F(x,y) = (-1,0)` anf the equation of the directrix is
`x = 1`, respectively.

Explanation:

Let be
`y^(2) = -4\cdot x`, which represents a parabola with a horizontal axis of symmetry. To find the location of the focus and to determine the equation of the directrix we must find the distance between focus and vertex (
`p`), dimensionless, a value than can be extracted from the following definition:

`(y-k)^(2) = 4\cdot p \cdot (x-h)` (Eq. 1)

Where:

`x` - Independent variable, dimensionless.

`y` - Dependent variable, dimensionless.

`h`,
`k` - Coordinates of the vertex, dimensionless.

By direct comparison, we get the following coincidences:

`h = 0`

`k = 0`

`4\cdot p = -4` (Eq. 2)

From (Eq. 2) we get that distance between focus and vertex is:

`p = -1`

Which is consistent with the fact parabola has an absolute maximum.

Now, the location of the focus is:

`F(x,y) = (h+p, k)` (Eq. 3)

If know that
`h = 0`,
`k = 0` and
`p = -1`, coordinates of the focus is:

`F(x, y) = (0-1, 0)`

`F(x,y) = (-1,0)`

And the equation of the directrix is represented by:

`x = -p` (Eq. 4)

(
`p = -1`)

`x = 1`

The coordinates of the focus are
`F(x,y) = (-1,0)` anf the equation of the directrix is
`x = 1`, respectively.