find the equation of the parabola

Question

with vertex (0,0) and focus (-2,0)?

Answer 1

The distance between the vertex and the focus is p.
p = sqrt [ ((-2-0)^2 +(0-0)^2] = sqrt(4) = 2
The focus lies below the vertex, so p=-2 and the parabola slopes downward.

The vertex and focus lie on the line x=0
Therefore, the parabola is of the form (x-h)^2 = 4p(y-k) , (h,k) being the vertx
(y-0)^2 = 4p(x-0)
y^2=-8(y-0)
y^2 = (-1/8) x

Answer 2

The vertex (0,0) and focus (-2,0)

Since the y coordinate of the vertex and focus is same .

So this is horizontal parabola where y  part is squred.

So the parabola opens leftward.

p  is distance between vertex and focus and also equals to distance between vertex and directrix.

p  = - 2 - 0 = -2

p  is negitive then this is horizontal parabola opens leftward.

The standard form of  parabola equation is  (y -k )² = 4p (x - h )

where vertex of parabola (h ,k ) and directrix is x = h - p

x = 0 - (-2)

Directrix is x = 2

Substitute the values (h, k ) = (0,0) and 4p = -8 in (y -k )² = 4p (x - h ).

(y - 0 )² = -8(x - 0)

y ² = -8x

The parabola equation is y² = -8x.