59. Form the equation of the geometric locus of points that are equally distant from the $O x$ axis and from the point $F(0 ; 2)$. Sketch the curve.
Solution. Let $M(x ; y)$ be a moving point describing the curve (coordinates $x$ and $y$ are current). Then, according to the problem, the distances from point $M$ to the $O x$ axis and to point $F$ are the same (Fig. $10, a$), i.e., $A M=F M$ or

$$
y=\sqrt{x^{2}+(y-2)^{2}}
$$

from which $y^{2}=x^{2}+(y-2)^{2}$.

We will transform the obtained equation to the form $y=f(x)$, i.e., solve it with respect to $y$. Finally, we will have

$$
y=\frac{x^{2}}{4}+1
$$

This is the equation of a parabola. Let's plot this curve. We will assign various values to $x$ and find the corresponding values of $y$ from the equation. We will obtain several pairs of values $(x ; y)$. The results of the calculations will be recorded in the form of a table $(10,8)$.

Now we will plot the points in the rectangular coordinate system corresponding to the pairs of numbers in the table and connect them with a smooth curve (Fig. $10, \sigma$).