Question

Derive the equation of the parabola with a focus at (0, 1) and a directrix of y = -1.c

Answer 1

`\displaystyle y=(1)/(4)x^2`

Explanation:

Let (x, y) be a point on the parabola.

By definition, any point on the parabola is equidistant from the focus and the directrix

The distance from the focus is given by:

`\begin{aligned} d&=\sqrt{(x-0)^2+(y-1)^2\\\\&=√(x^2+(y-1)^2)\end{aligned}`

The distance from the directrix is given by:

`d=|y-(-1)|=|y+1|\text{ or } |-1-y|`

So:

`√(x^2+(y-1)^2)=|y+1|^`

Square both sides. Since anything squared is positive, we can remove the absolute value:

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

Square:

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

Hence:

`x^2=4y`

So, our equation is:

`\displaystyle y=(1)/(4)x^2`