What is the standard form of the parabola with a vertex at (1,-2) and a focus at (1,3)?

1 Answer
Jul 24, 2018

The equation of the parabola is #y+2=1/20(x-1)^2#

Explanation:

The focus is #F=(1,3)# and the vertex is #V=(1,-2)#

Therefore, the directrix is #y=-7# as the vertex is the midpoint from the focus and the directrix

#(y+3)/2=-2#

#=>#, #y+3=-4#

#=>#, #y=-7#

Any point #(x,y)# on the parabola is equidistant from the focus and the directrix

#y+7=sqrt((x-1)^2+(y-3)^2)#

Squaring both sides

#(y+7)^2=(x-1)^2+(y-3)^2#

#y^2+14y+49=(x-1)^2+y^2-6y+9#

#20y+40=(x-1)^2#

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

#y+2=1/20(x-1)^2#

The equation of the parabola is #y+2=1/20(x-1)^2#

graph{(y+2-1/20(x-1)^2)(y+7)=0 [-16.02, 16.02, -8.01, 8.01]}