How do you find the standard form of the equation Vertex: ( -1, 2), Focus (-1, 0) Vertex: (-2, 1), Directrix: x = 1?

1 Answer
Oct 2, 2017

Answer:

Use the fact that a parabola is the locus of points equidistant from the focus point and the directrix line.

Explanation:

The distance from the directrix, #x =1#, to any point, #(x,y)#, on the parabola is:

#d = x - 1" [1]"#

The distance from the focus, #(-1,0)# to any point, #(x,y)#, on the parabola is:

#d = sqrt((x-(-1))^2+(y-0)^2)#

Simplify:

#d = sqrt((x+1)^2+y^2)" [2]"#

Because the distances must be equal, we can set the right side of equation [1] equal to the right side of equation [2]:

#x -1 = sqrt((x+1)^2+y^2)#

Square both sides:

#(x -1)^2 = (x+1)^2+y^2#

Expand the squares:

#x^2-2x+1 = x^2+2x+1+y^2#

Combine like terms:

#-4x = y^2#

Divide both sides by -4:

#x = -1/4y^2 larr# standard form for a parabola that opens left.