Question

A parabola can be represented by the equation y2 = –x. What are the coordinates of the focus and the equation of the directrix?

Answer 1

(y-k)²=4p(x-h)
(h,k) is vertex
and since we got the y term squred, it is facing left or right
when p is positive, then focus is to right of veretx
p is distance from focus to vertex and from vertex to dirextix

y²=-x
(y-0)²=4(-1/4)(x-0)
vertex is (0,0)
p is negative so then focus to left of vertex
so the focus is (-1/4,0)
dirextix is x=1/4

Answer 2

Answer:

The coordinates of the focus and equation of the directrix is $(-\frac{1}{4}, 0)$ and $x =\frac{1}{4}$ respectively.

Step-by-step explanation:

Consider the provided equation.

$y^2 = -x$

Rewrite the provided equation.

$(y-0)^2 = 4\times\frac{-1}{4}(x-0)$

The standard equation of parabola is $(y - k)^2 = 4p (x - h)$ where the focus is $(h + p, k)$ and the directrix is $x = h - p$.

By comparison we get:

Vertex is (0,0) and $p=-\frac{1}{4}$

Focus is:$(0 + \frac{-1}{4}, 0)$

Focus is:$(-\frac{1}{4}, 0)$

Directrix $x = 0 - \frac{-1}{4}$

Directrix $x =\frac{1}{4}$

Hence, the coordinates of the focus and equation of the directrix is $(-\frac{1}{4}, 0)$ and $x =\frac{1}{4}$ respectively.