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 .

Let be [tex]y^{2} = -4\cdot x[/tex], 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 ([tex]p[/tex]), dimensionless, a value than can be extracted from the following definition:

[tex](y-k)^{2} = 4\cdot p \cdot (x-h)[/tex] (Eq. 1)

Where:

[tex]x[/tex] - Independent variable, dimensionless.

[tex]y[/tex] - Dependent variable, dimensionless.

[tex]h[/tex], [tex]k[/tex] - Coordinates of the vertex, dimensionless.

By direct comparison, we get the following coincidences:

[tex]h = 0[/tex]

[tex]k = 0[/tex]

[tex]4\cdot p = -4[/tex] (Eq. 2)

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

[tex]p = -1[/tex]

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

Now, the location of the focus is:

[tex]F(x,y) = (h+p, k)[/tex] (Eq. 3)

If know that [tex]h = 0[/tex], [tex]k = 0[/tex] and [tex]p = -1[/tex], coordinates of the focus is:

[tex]F(x, y) = (0-1, 0)[/tex]

[tex]F(x,y) = (-1,0)[/tex]

And the equation of the directrix is represented by:

[tex]x = -p[/tex] (Eq. 4)

([tex]p = -1[/tex])

[tex]x = 1[/tex]

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

nick715 nick715 · Answer 2 · 2020-11-04T02:45:26+01:00

nick715 nick715

04-11-2020

Answer:

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

.

Step-by-step explanation:

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