(1) The standard form for a parabola opening left/right is `y^2=4px` if the vertex is at the origin. The directrix is the line x=p, and the focus is at (p,0)
Given `x+y^2=0` we can rewrite as `y^2=-x` or in standard form `y^2=4(-1/4)x` .(Set 4p=-1 to solve for p)
This is a parabola with vertex at the origin. Sine the coefficient is negative it opens to the left. The directrix will be p units to the right of the vertex, while the focus will be p units to the left of the vertex, along the line of symmetry.
The line of symmetry is the x-axis so we have vertex(0,0), focus `(-1/4,0)` , and directrix `x=1/4`
(2) The standard form for a parabola opening horizontally with vertex (h,k) is `(y-k)^2=4p(x-h)`
Given `4x-y^2-2y-33=0` we isolate the y terms:
`y^2+2y=4x-33` complete the square on the left-hand side
This is in standard form: the vertex is at (8,-1), the parabola opens right since p=1>0, the focus is 1 unit right of the vertex along the axis of symmetry (y=-1), and the directrix is the line x=7.
Thus the vertex (8,-1), focus (9,-1), directrix x=7
(3) The standard form of a parabola opening vertically with vertex(h,k) is `(x-h)^2=4p(y-k)`
Given `x^2-2x+8y+9=0` we isolate the x terms:
`x^2-2x=-8y-9` complete the square on the left hand side:
Thus p=-2 and we have:
The vertex is at (1,-1); the parabola opens down; the focus is at (1,-3) and the directrix is the line y=1