How do you find the equation of a parabola with directrix y=-1 and focus F(0,0)?

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Answer 1 · 2017-03-08T08:41:35

The equation of a parabola is #x^2-2y-1=0#

Parabola is the locus of a point, which moves so that its distance from a line, called directix, and a point, called focus, are equal.

We know that the distance between two points #(x_1,y_1)# and #x_2,y_2)# is given by #sqrt((x_2-x_1)^2+(y_2-y_1)^2)# and

the distance between point #(x_1,y_1)# and line #ax+by+c=0# is #|ax_1+by_1+c|/(sqrt(a^2+b^2)#.

Now distance of a point #(x,y)# on parabola from focus at #(0,0)# is #sqrt(x^2+y^2)#

and its distance from directrix #y=-1# or #y+1=0# is #|y+1|/sqrt(0^2+1^2)=|y+1|#

Hence, equation of parabola would be

#sqrt(x^2+y^2)=|y+1|# or

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

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

#x^2-2y-1=0#

graph{x^2-2y-1=0 [-2.592, 2.408, -1.07, 1.43]}