Parabola as a locus from focus and directrix

Text

Find the equation of the locus of points $P(x,y)$ equidistant from the focus $F(0,2)$ and from the directrix line $y=-2$.

Solution

Impose equality between the distance of $P$ from the focus and its distance from the directrix: $\sqrt{x^{2}+(y-2)^{2}}=|y+2|.$ Square both sides: $x^{2}+(y-2)^{2}=(y+2)^{2}\;\Rightarrow\; x^{2}+y^{2}-4y+4=y^{2}+4y+4.$ Cancelling $y^{2}$ and $4$: $x^{2}-4y=4y$, i.e. $x^{2}=8y$. The locus is the parabola with vertex at the origin $\boxed{\;y=\dfrac{x^{2}}{8}\;}$