Question

Use the Distance Formula to find the equation of a parabola with the given focus and directrix . F(0,1) Y=-1

Answer 1

For any point on the graph of a parabola, the distance to the focus and the distance to the directrix are equal.

The distance to the focus:

`(x,y) \\ F(0,1) \\ \\ d_1=√((x_2-x_1)^2+(y_2-y_1)^2)=√((0-x)^2+(1-y)^2)=\\=√((-x)^2+(1-y)^2)=√(x^2+1-2y+y^2)`

The distance to the directrix:
The directrix is a horizontal line, so the distance is the absolute value of the difference of the y-coordinates of the points.

`(x,y) \\ y=-1 \hbox{ so any point: } (m,-1) \\ \\ d_2=|y-(-1)|=|y+1|`

The distances are equal:

`d_1=d_2 \\ √(x^2+1-2y+y^2)=|y+1| \\ (√((x^2+1-2y+y^2))^2=|y+1|^2 \\ x^2+1-2y+y^2=(y+1)^2 \\ x^2+1-2y+y^2=y^2+2y+1 \\ x^2+1-1=y^2-y^2+2y+2y \\ x^2=4y \\ \boxed{y=(1)/(4)x^2}`