Question #46d00

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Answer 1 · 2017-10-30T18:12:47

The distance between any point, #(x,y)# on the parabola and the focus #(0,1)# is:

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

The distance any point, #(x,y)#, on the parabola and the line #y = -1# is:

#d = sqrt((y - (-1))^2)#

Because the definition of the parabola requires that these two distances be equal, we can set the right sides of both equations equal:

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

Square both sides of the equation and convert the -- to a +:

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

Expand the squares:

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

Combine like terms:

#4y = x^2#

Divide both sides by 4:

#y = 1/4x^2#

The derivation is complete.