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

Answer:

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

Step-by-step 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.